UNIVERSITY  OF  CALIFORNIA 
AT  LOS  ANGELES 


GIFT  OF 

MRS. JOHN   C.SHEDD 


Copyright,  1905, 
By  William  Francis  Magie. 


\:) 
• 


0/C31 


to 

INTRODUCTION. 

vS> 

1.  Subject-matter  of  Physics. — The  subject  of  physics  deals 

with  certain  phenomena  exhibited  by  bodies  in  the  material 
world.  Many  of  these  phenomena  have  been  classified  and 
described  under  general  laws,  which  are  expressed  in  mathe- 
matical form  and  can  be  made  the  basis  of  mathematical  dis- 
cussion. Other  phenomena,  however,  which  have  not  yet  thus 
been  classified,  properly  belong  to  the  subject  of  physics. 
Broadly  speaking,  those  natural  phenomena  which  either  have 
been  classified  or  which  we  expect  to  classify  in  this  way,  come 
under  the  general  subject  of  physics,  and  are  thus  distinguished 
,from  those  which  are  simply  classified  by  common  characteristics 
which  cannot  be  given  mathematical  formulation. 

2.  Method. — The  mental  process  involved  in  the  study  of 
physics  begins  with  the  data  of  experience  and  ends  with  the 
classification  or  description  of  a  body  of  phenomena,  expressed 
by  a  law  that  may  be  stated  in  mathematical  form.    The  first 
step  in  the  process  is  the  gathering  of  material  by  the  ob- 

<^J  servation  of  natural  phenomena.  Our  observations  are  some- 
__  times  made  as  the  phenomena  are  presented  to  us  in  opera- 
tions occurring  in  nature  and  not  subject  to  our  control;  but 
ordinarily  they  are  obtained  by  experimenting  upon  natural 
bodies  in  such  a  way  as  to  suppress,  so  far  as  possible,  those 
phenomena  which  are  not  immediately  the  object  of  observa- 
tion, and  to  make  prominent  those  which  we  desire  to  observe. 
The  results  of  these  experiments  are  considered  and  an  at- 
tempt is  made  to  discover,  in  the  phenomena  presented  by 
them,  some  general  or  simple  numerical  relation  from  which 
the  existence  of  a  general  rule  or  law  may  be  inferred. 

If  such  a  law  is  suspected,  it  is  then  put  in  mathematical 
form,  and  by  purely  logical  processes,  taking  it  in  connection 
with  other  known  laws  of  physics,  deductions  are  made  from 
it,  which,  when  interpreted  as  statements  of  fact,  express  re- 
sults which  may  be  examined  by  experiment. 


206520 


4  INTRODUCTION. 

Experiments  are  then  instituted  in  order  to  see  whether 
the  results  reached  by  the  argument  just  explained  have  their 
counterparts  in  nature.  If  it  is  found  that  properly, arranged 
experiments  exhibit  phenomena  agreeing  with  these  predicted 
results,  the  hypothetical  law  from  which  those  results  were 
obtained  is  confirmed.  The  amount  of  verification  of  this  sort 
which  an  hypothesis  requires,  before  its  confirmation' is  fully 
admitted  and  it  is  called  a  physical  law,  differs  in  different 
cases.  No  physical  law  can  be  confirmed  by  experiment  so 
completely  as  to  force  the  mind  to  accept  it  in  the  same  way 
that  the  mind  is  forced  to  accept  the  propositions  of  geometry 
as  necessary  consequences  of  the  axioms  and  postulates;  but 
as  continued  experiment  reveals  more  phenomena  which  arc 
necessary  results  of  the  hypothesis,  and  no  phenomena  which 
are  inconsistent  with  the  hypothesis,  our  belief  in  the  law  be- 
comes stronger.  Many  physical  laws  have  been  confirmed  so 
frequently  that  our  belief  in  their  truth  is  just  as  strong, 
although  based  on  other  grounds,  as  our  belief  in  the  con- 
clusions of  geometry. 

3.  Measurement. — In  order  that  the  physical  phenomena 
which  are  observed  can  be  connected  by  a  law  expressed  in 
mathematical  form,  they  must  be  measured,  Measurement 
consists  in  comparison  of  a  quantity  with  a  unit  of  its  own 
kind.  These  units  may  be  chosen  arbitrarily,  or  at  will,  but 
it  is  found  more  convenient  to  limit  the  arbitrary  choice  of 
units  to  a  few  of  them  which  are  considered  fundamental,  and 
to  develop  the  other  units  by  definition.  Those  units  which 
are  usually  chosen  as  fundamental  are  the  units  of  length,  of 
time,  and  of  mass.  The  unit  of  length  may  be  the  ordinary 
unit,  such  as  the  yard  or  the  metre,  which,  is  used  in  the  com- 
mon transactions  of  business.  In  physics,  however,  by  a 
convention  which  is  almost  universally  followed,  the  unit  of 
length  is  the  centimetre.  The  unit  of  time  may  be  any  unit 
in  which  intervals  of  time  are  measured:  in  physics  it  is 
commonly  the  second.  The  unit  of  mass  will  be  defined  when 
the  concept  of  mass  has  been  introduced  and  explained. 


MECHANICS. 

4.    Space  Relations  of  Forces. — The  first  important  scien- 
tific work  in  physics  was  done  by  the  Greek  geometer  Archi-     -n 
medes    (287-212  B.  C.).     The  Greek  mathematicians  had  de-     " 
voted  almost  all  their  attention  to  the  study  of  space  rela- 
tions, and  in  their  hands  geometry  had  attained  a  high  de- 
velopment.    Archimedes,   reflecting  on   the  concept  of   force, 
undertook  an  investigation  of  the  space  relations  of  forces. 

When  a  man  exerts  his  muscles  in  moving  a  body  or  in 
pushing  or  pulling  a  body  which  he  cannot  move,  he  is  con- 
scious of  the  exertion  of  force.  In  many  cases,  as  when  he 
bends  a  spring  or  sustains  a  weight,  he  interprets  his  sensa- 
tion by  saying,  that  the  body  against  which  he  acts  exerts  a  ^ 
force  in  the  opposite  direction  to  the  one  which  he  is  exerting 
and  of  equal  magnitude.  He  further  believes  that  the  body, 
when  in  the  same  position  or  condition,  is  still  exerting  a 
force,  even  though  he  does  not  directly  perceive  the  counter- 
acting force.  He  thus  feels  willing  to  use  bodies  in  certain 
conditions  or  positions  as  the  means  by  which  forces  may  be 
exerted. 

When  a  body  is  sustained  in  the  air  by  the  hand,  the  force 
exerted  by  the  hand,  as  is  shown  by  universal  experience,  is 
directed  vertically  upwards.  The  force  which  the  body  is  •  v .,.  . 
assumed  to  exert  in  the  opposite  direction  or  vertically  down-  ; 
wards,  is  called  its  weight.  Universal  experience  has  shown 
that  the  weight  of  a  body  depends  on  its  size  and  on  the 
material  of  which  it  is  composed.  If  different  bodies  com- 
posed of  the  same  material  and  of  equal  volume  are  tested  by 
the  hand,  or  in  any  other  way,  it  is  found  that  their  weights 
are  equal.  Two  of  these  bodies  will  have  the  same  weight  as 
a  body  of  the  same  material  and  of  a  volume  double  that  of 
either  one  of  them.  By  suitable-  combinations  of  equal 
weights,  any  one  of  which  is  selected  as  a  unit,  a  scale  of 

forces  can  be  constructed. 

• 


C  MECHANICS. 

If  a  body  is  supported  in  such  a  way  that  it  can  turn 
freely  in  a  vertical  plane  around  a  horizontal  axis,  at  right 
angles  to  that  plane,  and  is  then  acted  on  by  forces,  or  more 
specifically,  if  it  has  weights  hung  upon  it  at  different  points, 
it  will  generally  turn  about  the  axis  until  it  at  last  comes  to 
rest.  It  is  then  said  to  be  in  equilibrium.  It  is  natural  to 
suppose,  and  the  supposition  is  confirmed  by  common  observa- 
tion, that  this  condition  of  equilibrium  is  determined,  not 
merely  by  the  values  of  the  weights,  but  by  the  positions  of 
their  points  of  application.  The  problem  of  determining  the 
conditions  of  equilibrium  in  this  case  was  the  one  attacked 
by  Archimedes.  To  simplify  the  problem  as  much  as  possible, 
we  suppose  the  body  to  be  a  straight  rod,  itself  without 
weight,  so  that  it  can  be  treated  as  a  line  turning  on  an  axis 
passing  through  one  point  in  that  line.  Such  a  body  is  called 
an  ideal  lever.  Weights  are  suspended  at  different  points  on 
this  lever  and  are  shifted  about  until  the  lever  is  in  equi- 
librium. The  problem  then  is  to  determine  the  relation  which 
exists  between  the  weights  and  their  distances  from  the  axis, 
when  equilibrium  is  established.  Modern  analysis  of  the 
argument  of  Archimedes  has  shown  that  it  is  not  possible  to 
solve  this  problem  from  abstract  principles  by  an  argument 
which  does  not  somewhere  involve  the  very  principle  which  it 
is  desired  to  prove.  It  is  therefore  sufficient  to  investigate 
the  conditions  of  equilibrium  by  experiment. 

As  a  result  of  experiment  we  conclude  that  equilibrium 
will   exist  when  the  weights  which   tend   to  turn   the  lever 
around  the  axis  in  one  sense  are  so  placed  as  to  balance,  or 
counteract,  the  weights  which  tend  to  turn  the  lever  in  the 
opposite  sense.     Furthermore,  we  conclude  that  the  effect  of 
each  weight  depends  on  the  distance  of  its  point  of  applica- 
IY£<*I     tion  from  the  axis-     lf  we  multiply  each  weight  by  the  dis- 
'  tance  of  its  point  of  application  from  the  axis  and  collect  the 

j  ,\  V       Product9  in  two  groups,  according  as  the  weights  tend  to  turn 
\  the  lever  in  one  sense  or  the  other,  we  find  that,  in  the  case  of 

equilibrium,  the  sum  of  these  products  in  the  one  group  is 
equal  to  the  sum  in  the  other  group. 


MECHANICS.  7 

5.  Moments  of  Force. — If  the  lever  is  not  straight,  but 
bent,  so  that  the  points  of  application  of  the  weights  are  not 
in  the  same  straight  line,  the  rule  just  stated  does  not  hold 
true.    Experiment  shows,  in  this  case,  that  the  importance  of 

any  weight  in  determining  equilibrium  does  not  depend  on  the    ~f  .     ^ 

product  of  the  weight  and  of  the  distance  of  its  point  of  ap-          .4 

plication  from  the  axis,  but  on  the  product  of  the  weight  and 

of  the  distance  from   the  axis   to   the  vertical   line   pajsing 

through  the  point  of  application  of  the  weight.     This  product 

is  called  the  moment  of  the  weight,  or,  if  we  take  the  weight 

simply  as  a  representative  force,  it  may  be  called  thejnoment 

offeree.     The  condition  of  equilibrium,  in  the  case  of  a  lever 

oT  any~  form,  may  then  be  stated  by  saying,  that  equilibrium 

w.ill  exist  when  the  sum  of  the  moments j?l  force,  which  tend  to 

turn  the  lever  in  one  sense  equals  the  sum  of  the  moments  of 

force  which  tend  to  turn  it  in  the  opposite  aense.     If  we  give 

to  those  moments  of  force,  which  tend  to  turn  the  lever  In 

one  sense,  the  positr£e_sign,  and  to  those  which  tend  to  turn 

it  in  the  opposite  sense,  the  negative  sign,  the  condition  of 

equilibrium  may  be  stated  by  saying,  that  the  algebraic  sum 

of  the  moments  of  force  which  act  on  the  lever  is  equal  to 

zero. 

6.  Resultant  of  Parallel  Forces. — Experiment  shows  that 
when  a  lever  is  in  equilibrium,  it  may  be  sustained  by  a  force 

equal  to  the  sum  of  the  weights  applied  to  the  lever.     The  ^.V  (^ 
single  force  which  is  equal  to  this,  and  so  is  equal  to  the  sum          ' 
of  the  weights,  is  called  the  resultant  of  the  weights.     By  a 
generalization   which   is   justified   by   experience,   any    set  of 
parallel  forces  which  are  applied  to  a  body  so  as  to  produce 
equilibrium  about  an  axis,  which  is  kept  fixed  by  an  opposite 
force,  is  said  to  have  a  resultant  equal  to  the  sum  of  the 
parallel  forces. 

When  the  axis  is  kept  fixed  by  means  of  rigid  connections 
with  bodies  which  are  not  perceived  to  be  exerting  force, 
although  they  may  be  doing  so,  the  axis  is  called  the  fulcrum 
of  the  lever.  The  experiments  just  described  show  that  we 
may  substitute  for  the  fulcrum  a  force  equal  to  the  resultant 


MECHANICS. 


of  the' parallel  forces  acting  on  the  lever,  without  disturbing 
the  conditions  of  equilibrium. 

7.  Centre  of  Gravity.— If  an  extended  body  is  hung  from  a 
flexible  thread  attached  to  a  point   in  it,  it  will  take  up  a 
position  of  equilibrium.     In  this  position  it  is  plain  that  the 
moments  of  force  due  to  the  weights  of  the  different  parts  of 
the  body  balance  each  other  around  any  horizontal  axis  which 
passes  through  the  vertical  line  determined  by  the  thread.    If 
the  body  is  then  hung  from  another  point  of_it,  so  that  it  is 
again  in  equilibrium,  the  moments  of  force  due  to  the  several 
parts  will   manifestly   conform  to   the   same   condition.     Ex- 
periment indicates,  and  analysis  proves,  that  any_two   lines 
thus  determined  will  cut  each  other  at  a  definite  point.     If  a 
horizontal  axis  is  passed  through  this  point  in  any  direction 
in  the  body,  the  body  will  then  be  in  equilibrium. about  that 
axis.     Or,'  if  a  fixed  support  is  placed  at  that  point,  the  body 
will  be  in  equilibrium  on  that  support,  whatever  be  its  posi- 

jtion.      The    point    thus    determined    is    called    the^ccntre    of 
•  gravity.     It  is  the  point  at  which  we  may  consider  the  re- 
Isultant  force  applied  which  is  equal  to  the  sum  of  the  weights 
of  the  different  parts  of  the  body.     When  we   speak  of  the 
weight  of  a  body  as  a  force,  we  commonly  mean  the  resultant 
force  thus  determined,  and  its  point  of  application  is  some- 
where  on    the   vertical    line    passing    through    the    centre   of 
gravity. 

8.  The  Lever. — In  the  common  use  of  the  lever,  the  lever 
itself  is  sustained  by  a  fulcrum  and  two  \veights  are  applied 
to  it,  one  of  which  is  called  the  power  and  the  other  the 
weight.     The  ratio  of  the  weight  to  the  power  is  called  the 
mechanical  advantage  of  the  lever.     The  distances  from  the 
axis  passing  through  the  fulcrum  to  the  vertical  lines  passing 
through   the  points  of  application  of  the  power  and   of  the 
weight  respectively   are  called   the   arms   of   the   lever.     The 
general  law  of  equilibrium  may  £hen  be  stated  by  saying  that 
the  lever  is  in  equilibrium  when  the  power  is  to  the  weight  in 
the  inverse  ratio  of  their  respective  lever  arms;  or  when  the 
jHQdjJct   of   the   power^  and   the   power   arm    is   equal    to   the 
product  of  the  wejght^  and  the  weight  arm. 


MECHANICS. 

The  different  sorts  of  levers  of  this  simple  type  may  best 
be  described  by  substituting  for  the  fulcrum  a  force  equal  to 
the  sum  of  the  power  and  the  weight  and  applied  to  the  axis 
vertically  upward.  These  three  forces  form  a  system  of 
parallel  forces  in  equilibrium,  for  any  one  of  which  we  may 
substitute  a  fulcrum  without  altering  the  equilibrium  of  the 
two  remaining  forces.  If  we  designate  one  of  the  two  forces  j- *~C  ^ 
originally  applied  to  the  lever  by  X,  the  other  by  Y,  and  the  ~sy  ^ 

third  force  or  resultant  by  Z,  we  may  describe  the  different  ^  \ 

sorts  of  levers  as  follows:    The  lever  is  of  the  first  kind  when     (/  '   V  \ 

cither  Xj3r  Y  is  the  power,  the  other  being  the  weighjl     The  . 

lever  is  oTthe  second  Jdnd  when  X_or_Y  is  the  power  and  Z  is   s-\     * 
the  weight.     The  lever"  is  of  the  third  kind  when__Z   is  the  ^ 
power  and  X  or  Y  is  the  weight.     In  some  of  these  cases  the      ,-*. 
mechanical  advantage  is  greater  than   1,  so  that  the  weight    (3j  ^ 
sustained,  or  the  force  exerted  which  takes  the  place  of  the  - 
weight,    is    greater    than    the    power.      In    other    cases    the 
mechanical  advantage  is  less  than   1  or  is  a  proper  fraction. 

9.    The  Principle  of  Work. — The  equality  of  moments  on        /  .      /  / 

both  sides  of  the  axis,  which  has  been  found  to  be  the  condi-    -7 **-*  /*~—  *^f^" 
tion  of  equilibrium,  may  seem  at  first  sight  to  be  a   sort  of<*-»»  -*«-«^ 
artificial   or  at  least  accidental   condition.     The   weights   are 
different  and  the  lever  arms  are  different.    The  fact  that  their 
respective  products  are  equal  does  not  of  itself  indicate  any- 
thing  similar  to  the  equality  between  cause  and  effect  which 
is  accepted  so  commonly  as  a  fundamental  principle  of  reason- 
ing.   The  condition  of  equilibrium,  however,  may  be  expressed 
in  another   form,   in   which   this  equality   between  cause   and 
effect  is  brought  out. 

Suppose  the  lever  in  equilibrium  under  the  action  of  two    ^/^  7 
forces  or   weights,   and   suppose   it   to   turn   through   a   small  v 

angle  about  the  fulcrum.  To  simplify  the  conditions  as  much 
as  possible  we  suppose  the  lever  straight  and  horizontal. 
Then  if  it  turns  through  the  small  angle  o,  one  of  the  forces 
will  be  lifted  through  a  distance  which  may  be  expressed  by 
pa  while  the  other  is  lowered  through  a  distance  expressed  by  •wo. 
If  we  multiply  these  distances  by  the  forces  applied  at  the  points 


JO  MECHANICS. 

which  move  through  these  distances,  we  find  from  the  principle  of 
moments  already  established,  that  the  two  products  Pp*  and  Ww* 
are  equal. 

Now  the  movement  of  the  point  of  application  of  a  force 
through  a  distance  measured  along  the  line  of  the  force  is  an 
effect  which  may  be  measured  by  the  product  of  the  force  and 
the  distance.  This  productjsjiamed  the  work  done  by  the 
force,  or  if  the  movement  of  the  point  of  application  is  in 
the  direction  opposite  to  that  of  the  force,  it  is  named  the 
work  done  against  the  force.  The  condition  of  equilibrium  in 
this  case  may  then  be  stated  by  saying  that  equilibrium 
exists  when,  for  a  small  displacement  of  the  lever  about  its 
axis,  the  work  done  by  the  one  force  is  equal  to  the  work  done 
against  the  other 

In  case  the  lever  is  bent,  a  similar  investigation  will  show 
that  the  lever  will  be  in  equilibrium  when  the  products  are 
equal  which  are  formed  by  multiplying  the  forces  by  the  re- 
spective vertical  distances  through  which  their  points  of  ap- 
plication rise  or  sink  above  or  below  their  original  positions. 
Either  of  these  products  is  named  the  work  done  by  the  force 
which_enters  in  it  as  a^Eaclor.  In  so  denning  the  product  we 
use  the  convention  that  work  done  against  the  force  may  be 
considered  as  negative  work  done  by  the  force.  So,  in  gen- 
eral, any  lever  will  be  in  equijikrjuin  when,  for  a  small  dis- 
placement about  the  fulcrum,  the  positive  work  done  by 
some  of  the  forces  which  act  on  it  is  numerically  equal  to 
the  negative  work  done  J>y  the  remaining  forces,  or,  when  the 
sum  of  all  the  work  done  by  all  the  forces  is  equal  to  zero. 

The  condition  of  equilibrium  may  otherwise  be  stated,  at 
least  when  the  forces  applied  to  it  are  weights,  by  saying  that 
for  any  small  displacement  of  the  lever  from  its  position  of 
equilibrium  the  centre_pf  gravitv_QL  the_jweights_j^iseSy  or  at 
least  does  not  sink.  That  this  statement  is  true  in  the  case 
of  the  simple  straight  lever  with  two  weights  may  be  seen  by 
inspection,  and  its  truth  in  more  complicated  cases  of  the  bent 
lever  and  of  many  weights  may  be  demonstrated  without 
difficulty.  It  is  of  interest  because  it  was  ajiopted  hotly  by_ 


MECHANICS.  11 

Galileo  and  by  Huygens  as  a  fundamental  principle,  the  evi- 
dence for  which  they  did  not  attempt  to  give.  In  their  minds 
its  validity  was  evident.  So  far  as  it  needed  demonstration, 
they  showed  that  its  denial  involved  the  denial  of  the  well 
known  truth  that  bodies  of  themselves  do  not  rise  or  move 
away  from  the  earth. 

When  we  are  considering  the  equilibrium  of  a  single  lever, 
the  principle  of  work  may  seem  to  be  little  more  than  a  mere 
restatement  of  the  principle  of  moments.  It  is  only  when  we 
are  dealing  with  systems  of  levers  that  the  peculiar  advan- 
tages of  the  principle  of  work  appear.  As  an  illustration,  let 
us  consider  two  levers  so  connected  that  the  weight  end  of 
the  one  is  applied  to  the  pqwer_end_Qf__the  other.  This  pair  -4  tV  & 
of  levers  will  be  in  equilibrium  when  a  force  is  applied  to  the  vj 
power  end  of  the  first  one  and  a  weight  of  a  certain  magnitude 
to  the  weight  end  of  the  second  one.  The  ratio  betweeji_this 
weight  and  the_j>ower  can  be  worked  out  by  the  aid  of  the 
principle  of  moments  applied  successively  to  the  two  levers, 
on  the  supposition  that,  when  the  first  lever  is  in  equilibrium, 
the  force  which  its  weight  end  applies  to  the  power  end  of 
the  second  lever  is  that  force  which  will  maintain  the  first 
lever  in  equilibrium  by  itself.  When  worked  out  by  the 
principle  of  moments  we  find  that  equilibrium  will  exist  when 
the  product  of  the  power  and  thp  pnwpr  arms  ia^egual  to  the 
product  of  the  weight  and  the  weight  arms.  In  determining 
the  conditions  of  equilibrium  of  this  system,  or  of  any  more 
complicated  system,  by  the  principle  of  moments,  our 
knowledge  of  the  several  levers  which  make  up  the  system  must 
be  complete. 

If  we  now  consider  the  work  done  when  the  system  of 
levers  undergoes  a  small  displacement,  we  perceive  that  the 
forces  which  are  exerted  between  two  levers  of  the  system  are 
equal  and  opposite  at  the  point  of  junction,  so  that  no  work 
is  done  when  that  point  moves.  The  same  being  true  for 
every  point  of  junction,  the  only  work  that  is  done  is  that 
done  by  the  forces  applied  to  the  two__frge  ends  of  the  system. 
Equilibrium  will  exist  when  the  work  done  by  one__oJL  these 


1:2  MECHANICS. 

forces  is  equal  and  opposite  in  sign  to  that  done  by  the  other 
force.  The  ratio  of  the  weight  to  the  power  is  therefore  the 
inverse  ratio  of  the  distances  through  which  the  weight  and 

I  the  power  move  respectively,  and  to  determine  the  conditions 
of  equilibrium,  no  knowledge  of  the  intermediate  structure 
of  the  system  is  necessary.  If  we  know_simjgily  the  vertical 
distances  through  which  the  two  ends  of  the  system  move, 
when  the  system  undergoes  a  small  displacement,  we  are  able 
to  determine  the  conditions  of  equilibrium. 

10.  The  Pulley. — The  pulley  is  a  wheel  turning  freely  on 
a  horizontal  axis.  Over  a  groove  in  the  rim  of  the  wheel  is 
passed  a  flexible  cord.  Weights  may  be  hung  on  the  two  ends 
of  the  cord  or  forces  may  be  applied  to  them.  If  the  supports 
of  the  axle  are  fixed,  the  pulley  is  called  a  fixed  pulley. 
Whether  the  two  sections  of  the  cord  are  parallel  or  not,  it 
is  plain  from  symmetry  that  the  fixed  pulley  will  be  in  equi- 
librium when  the  forces  applied  to  the  ends  of  the  cord  are 
equal.  This  condition  of  equilibrium  may  be  otherwise 
readily  deduced,  l)y  considering  Uie  points  on  the  wheel  at 
which  thg  cords  leave  it.  as  the  points  of  application^  the 
forces  which  are  applied  to  the  cord,  and  considering  the 
rjidii_of  the  wheel  drawn  from  the  axis  to  those  points  as  two 
level-  arms  of  equaMength.  The  moments  of  force  on  either 
side  of  the  axis  are  then  plainly  equal,  when  the  weights  or 
forces  are  equal. 

(The  mechanical  advantage  of  the  fixed  pullgy  is  1.  and  its 
"<e  is  simply  to  change"the  ^direction  of  thejiorcg  which  is 
applied  to  the  cord. 

Sometimes  the  pulley  is  not  fixed  on  a  support,  but  is  hung 
in  a  loop  of  the  cord,  one  end  of  which  is  fastened  to  a  fixed 
«.     Ct  support,  while  a  force  is  applied  to  the  other  end.     A  weight 

is  hung  on  the  frame  work  in  which  the  axle  of  the  pulley  is 
supported.  Such  a  pulley  is  called_a_jnovable  pulley.  The 
conditions  of  equilibrium,  that  is  the  relative  magnitudes  of 
this  weight  and  of  the  force  which  must  be  applied  to  the 
cord  at  its  free  end  to  sustain  the  weight,  may  be  deduced  by 
the  method,  already  employed,  of  tr^a^g^thejgullev^ as  an 


MECHANICS. 

vr 

equal  armed  lever.  To  simplify  our  conceptions,  we  suppose 
the  free  end  of  the  cord  is  to  be  pulled  vertically  upward  by 
a  force.  Then,  if  the  pulley  is  in  equilibrium,  we  may  sub- 
stitute for  the  fixed  support  another  force  equal  to  the  one 
applied  to  the  free  end  of  the  cord,  and  also  directed  vertically 
upward.  The  resultant  of  these  two  equal  forces,  and  there- 
fore the  weight  which  they  will  sustain,  is  equal  to  the  sum 
of  these  forces,  or  is  double  either  one  of  them.  The^me- 
chanical  advantage  of  the  movable  pulley  is  therefore  2. 

Combinations  of  fixed  and  movable  pulleys  may  be  made 
(e.   g.   the   block   and   tackle)    of   which   the   mechanical   ad- 
vantage is  greater  than  2.     In  typical  cases  of  this  sort  several 
pulleys  are  mounted  together  in  a  single  frame  work  or  block, 
and  two  branches  of  the  cord  are  associated  with  each  of  the 
pulleys    in   the   block.      The    mechanical    advantage    is    conse-  . 
quently  twice  the  number  of  movable__Dulj£ys,  or  is  equal  to/ 
the  number  of  branches  of  the  cord  passing  over  the  movable  \ 
pulleys! 

The  principle  of  work  applies  also  to  the  case  of  combina- 
tions of  fixed  and  movable  pulleys.  When  the  system  is  in 
equilibrium,  if  the  weight  hung  on  the  tree  end  of  the  cord  is 
lowered  through  a  certain  distance,  the  block  of  movable 
pulleys  is  lifted  through  a  distance  equal^  to  that  through 
which  the  aforesaid  weight  i^Jowered,  divided  by  the  number 
of  branches  of  the  cord  associated  with  the  movable  pulleys. 
Thus  if  the  applied  weight  is  lowered  through  the  distance  _s, 
and  there  are  2n  branches  of  the  cord,  the  distance  through 
which  the  weighT  attached  to  the  fhovable  pulleys  is  lifted  is 
s-r-2n.  The  work  done  on  the  weight  P  falling  through  the  dis- 
tance s  is  Ps.  The  work  done  on  the  weight  IF  as  it  is  lifted 
through  the  distance  s-=-2n  is  Ws-r-gn^  If  the  system  is  in 
equilibrium,  these  two  quantities  of  work  are  equal,  and  ..  /  «.  / 
hence  W=2nP,  which  is  the  relation  already  obtained  be-  _  /* 
tween  tnlTtwo  weights.  ^^ 

11.  Non-parallel  Forces. — It  has  been  shown  that  when  the 
forces  applied  to  a  lever  in  equilibrium  are  parallel,  the 
resultant  which  passes  through  the  fulcrum  is  equal  to  the 


]4  MECHANICS. 

sum  of  the  forces  which  are  applied  to  the  lever.  This  is  not 
thTjeastTif^the  forces  applied  to  the  lever  are  ngt  parallel. 
Equilibrium  may  still  exist  when  the  forces  are  not  parallel, 
andTnT~condition  of  equilibrium  is  still  the  same,  namely, 
that  the  moments_ofjforce  on  either  side  of  the  axis  are  equal. 
But  the  resultant  of  the  non-parallel  forces,  though  it  passes 
through  the  fulcrum,  is  not  equal  to  the  sum  of  the  forces. 

The  simplest  way  in  which  we  can  investigate  the  relation 
of  the  resultant  of  non-parallel  forces  to  those  forces,  is  to 
examine  by  experiment  the  case  of  two  forces  applied  at  one 
point  in  different  directions.  The  point  to  which  these  forces 
are  applied  may  be  kept  in  equilibrium  by  the  application  at 
the  same  point  of  a  force  properly  determined  in  magnitude 
and  direction.  In  the  special  cases  which  we  can  examine  we 
find  that  this  third  force,  which  is  plainly  equal  and  opposite 
to  the  resultant  of  the  two  given  forces,  may  be  found  by  the 
following  rule:  From  any  point  as  origin  draw  two  lines  in  ' 
the  directions  of  the  given  forces,  lay  off  on  those  lines  lengths 
proportional  to  the  magnitudes  of  the  forces,  and  with  those 
lengths  as  sides,  complete  a  parallelogram  and  draw  its  diag- 
onal from  the  origin.  The  direction  of  this  diagonal  will  be 
the  direction  of  the  resultant  and  the  length  of  the  diagonal 
will  be  proportional  to  the  magnitude  of  the  resultant.  This 

I  rule  is  called  the  parallelogram  of  forces.  Special  cases  of  it 
were  cited  by  Aristotle  (384-322  B.  C.).  It  was  first  clearly 
stated  by  Newton  (1687  A.  D.).  The  way  in  which  Newton 
developed  it  will  be  given  later. 

It  may  be  of  interest  to  examine  the  way  in  which  this 
parallelogram  law  for  obtaining  the  resultant  of  two  forces 
which  are  not  parallel  may  be  deduced  from  the  principle  of 
moments. 

We  consider  the  case  of  twojorces,  not  parallel,  applied  to 
ajlever  and  in  equilibrium.  It  is  plain  TKatTthe  resultant  of 
these  forces  musfpass  through  the  fulcrum.  The  problem  is 
to  find  the  direction  of  this  resultant  and  its  magnitude.  If 
the  lines  of  the  two  forces  are  prolonged  untiOhey  cross,  it  ia 
also  plain  that  the  resultant  will  be  in  a  line  which  passes 


MECHANICS. 


15 


through  the  point  of  intersection.  About  this  point  the 
moments  of  the  two  forces  and  of  the  resultant  are  zero.  If 
the  resultant  is  to  be  in  every  way  equivalent  to  the  two 
forces,  the  moment  of  the  resultant  about  any  other  point 
must  be  equal  to  the  sum  of  the  moments  of  the  two  forces 
about  the  same  point. 

From  the  point  P  draw  the  lines  PQ  and  PS  proportional  in 
length  to  the  given  forces.  From  the  point  S  draw  the  line 
ST  parallel  and  equal  to  the  line  PQ,  join  PT,  choose  any 
point  O  and  draw  the  lines  OP,  OQ,  OS,  OT.  The  triangles 
OPQ  and  OPS  are  proportional  to  the  moments  of  the  two 
forces  about  the  point  O.  Now  the  triangle  OPT  is  equal  to 
the  sum  of  the  other  two  triangles,  for  the  altitude  of  the 
triangle  OPT,  standing  on  the  base  OP,  i*  equal  to  the  sum 
of  the  altitudes  of  the  triangles  OPQ  and  OPS,  which  stand 
on  the  same  base.  The  moment  of  the  force  represented  by 
PT  is  therefore  equal  to  the  sum  of  the  i. moments  of  the  two 
given  forces.  And  it  may  easily  be  seen  that  no  other  position 
of  the  point  T  but  the  one  determined  by  the  construction 
already  given  will  make  the  altitude  of  *.he  triangle  OPT 
equal  to  the  sum  of  the  altitudes  of  the  other  two  triangles 
for  any  arbitrarily  chosen  position  of  the  point  O.  The 
resultant  PT  thus  determined,  which  is  *he  same  as  that 
determined  by  the  parallelogram  law,  is  therefore  the  only 
force  which  will  have  a  moment  about  any  point  equal  to  the 
sum  of  the  moments  of  the  two  given  forces,  and  it  is  there- 
fore the  only  force  which  is  equivalent  in  all  respects  to  the 
two  given  forces. 

The  parallelogram  law  can  be  deduced  in  this  way   from  | 
the  principle  of  moments.     If  we  accept  as  fundamental  the  • 
parallelogram^  law,    we    may,    by    a    demonstration    which    is 
essentially  similar,  deduce  from  it  the  principle  of  momenta. 

12.    Composition  and  Resolution  of  Forces. — As  was  shown    I 
in  the  last  section,  we  may  represent  a  force  by  a  line,  the    \ 
direction  of  the  line  representing  the  direction  of  the  force, 
and   the   number  of  units  of  length  of  the   line   representing 
the  magnitude  of  the  force  measured  in  terms  of  some  unit. 


h' 


16 


MECHANICS. 


This  mode  of  representation  of  forces  facilitates  the  study  of 
combinations  of  forces.  The  rule  for  finding  the  resultant  of 
two  forces  has  already  been  given.  If  more  than  two  forces 
are  applied  at  a  point,  their  resultant  can  be  found  by  using 
this  rule  with  two  of  the  forces  to  find  their  resultant,  using 
this  resultant  with  another  of  the  forces  to  obtain  J*__ngw^ 
resultant,  and  proceeding  in  this  way  until  all  the  forceshave 
been  used,  and  a  final  resultant  is  reached.  A  simpler  way 
of  obtaining  the  same  resultant  is  as  follows:  From  any 
point  as  origin,  draw  a  line  to  represent  one  of  the  forces; 
from  the  end  point  of  that  line  draw  a  line  to  represent  an- 
other of  the  forces;  from  the  end  point  of  that  line  draw  a 
line  to  represent  another  of  the  forces,  and  proceed  in  this 
way  until  all  the  forces  have  been  introduced.  The  result 
of  this  operation  is  a  broken  line,  which  may  be  made  the 
perimeter  of  a  polygon  by  drawing  a  line  from  the  origin  to 
the  end  point  of  the  line  representing  the  last  force  consid- 
ered. This  line  is  the  resultant  of  the  forces.  It  may  hap- 
pen that  the  last  point  of  the  line  representing  the  last  force 
considered  coincides  with  the  origin.  In  that  case  the  result- 
ant of  the  set  of  forces  is  zero,  or  the  forces  are  in  equi- 
librium. 

It  is  often  important  to  find  two  or  more  forces  which 
have  a  resultant  equal  to  a  given  force.  This  cannot  be  done 
unless  a  sufficient  number  of  conditions  are  given  to  which 
the  forces  to  be  found  shall  conform.  In  the  simplest  case, 
in  which  two  forces  are  to  be  found  whose  resultant  is  equal 
to  a  given  force,  two  conditions  determining  either  the  direc- 
tion or  the  magnitude  of  one  or  both  forces  must  be  given. 

I  The  process  of  finding  these  forces  is  called  the  resolution 
of  a  force.  The  forces  which  are  found  are  called  the  compo- 
nents of  the  given  force,  and  the  force  whose  components  are 
found  is  said  to  be  resolved  into  them. 

As  an  illustration  of  the  process  of  resolving  a  force  into 
two  components,  we  may  take  the  case  of  a  given  force  whose 
components  in  two  given  direction  are  desired.  From  any 
point  draw  a  line  representing  the  given  force;  from  the  same 


MECHANICS.  17 

point  draw  two  lines  of  indefinite  length  in  the  given  direc-  . 

tions,  which  must  be  on  either  side  of  the  direction  of  the  *s\    V 
given   force;    with  the   line   representing  the  given   force  as 
diagonal,  construct  the  parallelogram  whose  sides  are  in  the  I 
given  directions ;   the  components  desired  are  then  given  by  I 
the  sides  of  this  parallelogram. 

We  are  often  asked  to  find  the  component  of  a  force  in  a 
given  direction  or  making  a  given  angle  with  the  given  force. 
The  problem  as  thus  stated  lacks  one  of  the  two  necessary 
conditions  to  enable  us  to  reach  a  definite  result;   but  when 
it  is  stated  in  this  way,  it  is  always  assumed  that  the  other  \ 
component,  about  which  nothing  is  said,  is  to  be  taken  in  a  j 
direction  at  right  angles  to  the  one  which  is  desired.     With  | 
this  understanding,  the  required  component  can  be  found  by      . 
drawing  the  line  representing  the  given  force,  and  then  draw-   ••-/ 
ing  two  lines  from  the  same  origin,  one  of  which  is  in  the 
given  direction   and  the  other  at   right  angles  to   it.     When 
the   rectangular    parallelogram    is   constructed,   on   these    two 
directions,    about    the    line    representing    the    given    force    as 
diagonal,  the  side  of  the  parallelogram  which  is  in  the  given 
direction  is  the  component  desired.     If  the  angle  between  the 
direction  of  the  given  force  F  and  the  direction  of  the  desired 
component  is  a,  the  magnitude  of  the  component  is  F  cos  a. 

13.    The  Inclined  Plane. — If  a  perfectly  smooth  rigid  plane  1 
is  inclined  to  the  horizontal  at  any  angle,  a  weight  may  be  I 
supported  on  it  by  a  force  directed  upward  parallel  with  the 
surface  of  the  plane.    Such  a  plane  is  called  an  inclined  plane, 
and  it  is  a  fundamental  problem  in,  mechanics  to  find  the  me- 
chanical advantage  afforded  by  such  a  plane,  that  is,  to  find 
the  ratio  of  the  weight  on  the  plane  to  the  force  which  will 
sustain  it. 

An    interesting    solution    of    this    problem,   which    rests  I 
directly  upon  fundamental  perceptions,  and  does  not  involve    J 
the  use  of  any  formulated  principle  of  mechanics,  was  given   / 
by  Stevinus.     Stevinus'  solution  is  as  follows:     Let  us  sup- 
pose two  inclined  planes  such  as  the  one  which  has  been  de- 
scribed, starting  from  the  same  horizontal  plane,  and  meeting 


18 


at  a  summit.  Let  us  suppose,  further,  a  perfectly  smooth  and 
flexible,  heavy,  endless  cord  hung  over  the  planes.  Everyone 
will  admit  on  inspection  of  this  arrangement  that,  if  the  cord 
is  originally  at  rest,  it  will  remain  at  rest,  that  is,  its  weight 
will  be  so  distributed  that  the  cord  will  be  in  equilibrium.  Jf 
this  were  not  so,  the  cord  would  have  of  itself  the  power  of 
initiating  and  maintaining  continuous  motion  around  the  in- 
clined planes.  Since  the  part  of  the  cord  which  hangs  below 
the  lowest  points  of  the  planes  is  symmetrical  with  respect  to 
those  points,  the  condition  of  equilibrium  must  be  maintained 
by  those  parts  of  the  cord  which  rest  on  the  two  planes.  Since 
equilibrium  is  maintained  the  forces  which  these  two  parts 
exert  in  opposite  directions  are  equal.  Now  the  weights  of 
the  two  parts  are  proportional  to  their  lengths;  and  therefore 
the  weights  which  will  be  sustained  by  equal  forces  on  planes 
of  the  same  height  are  proportional  to  the  lengths  of  the 
planes. 

If  one  of  the  two  planes  is  perpendicular,  the  force  ex- 
erted by  the  part  of  the  cord  which  hangs  perpendicularly  is 
the  weight  of  that  part.  It  will  sustain  on  the  inclined  plane 
a  part  of  the  cord  equal  to  the  length  of  the  inclined  plane. 
The  weight  of  the  perpendicular  part  of  the  cord  is  equal  to 
the  force  which,  applied  parallel  to  the  surface  of  the  inclined 
plane,  will  sustain  on  it  the  inclined  part  of  the  cord.  The 
ratio  of  the  weight  of  the  inclined  part  of  the  cord  to  that 
force  is  the  mechanical  advantage  of  the  inclined  plane,  and 
it  is  therefore  equal_to^  the  ratio  of  the  length  of  the  plane  to 
its  height. 

Another  solution  of  the  same  problem  was  given  by  Galileo, 
which  is  based  on  the  principle  of  work.  Suppose  the  in- 
clined plane  constructed  and  that  a  weight  placed  upon  it  is 
sustained  by  a  cord,  parallel  with  the  surface  of  the  plane, 
passed  over  a  pulley  at  the  summit  of  the  plane,  and  attached 
to  another  weight  of  properly  chosen  magnitude,  which  hangs 
vertically.  Galileo  asserted  that  when  this  system  of  two 
weights  is  in  equilibrium,  a  small  displacement  of  the  system 
may  be  effected  without  doing  any  work.  Suppose  this  dis- 


MECHANICS. 


to 

hi.  / 

his  J 


placement  effected  by  raising  or  lowering  the  weight  which 
hangs  vertically;  then  the  work  done  by  that  weight  is  equal 
to  the  weight  multiplied  by  the  distance  through  which  it  is 
moved.  The  weight  which  stands  on  the  plane  is  at  the  same 
time  lowered  or  raised  along  the  plane  by  the  same  distance, 
and  is  lowered  or  raised  vertically  by  a  distance  which  is 
proportional  to  that  distance  in  the  ratio  *>r  the  height  of  the 
plane  to  its  length.  Since  the  work  done  by  the  weight  on 
the  plane  is  equal  to  that  weight  multiplied  by  the  vertical 
distance  through  which  it  moves,  in  order  chat  the  work  done 
by  the  weight  on  the  plane  shall  be  equal  to  the  work  done  by 
the  other  weight,  the  weight  on  the  plane  must  be  greater 
than  the  other  one  in  the  ratio  of  the  length  of  the  plane  to 
its  height.  The  same  conclusion  is  therefore  reached  by  this 
method  of  argument  as  was  reached  by  Stevinus  by  his 
method. 

We  may  use  this  result  as  a  basis  from  which  to  demon- 
strate the  parallelogram  law.  It  is,  however,  more  ad- 
vantageous to  assume  the  parallelogram  law,  and  by  its  help 
to  demonstrate  the  same  law  of  the  inclined  plane.  Suppose 
a  weight  to  rest  on  the  -plane  in  equilibrium.  The  weight  «y 
itself  is  a  force  directed  vertically  downward.  Resolve  it  into  * 
two  components,  one  perpendicular  to  the  plane,  the  other 
parallel  with  the  plane.  The  component  perpendicular  to 
the  plane  is  a  force  which  does  not  need  to  be  counteracted  by 
any  other  force  applied  to  the  weight.  The  component  par- 
allel to  the  plane  must  be  counteracted  by  an  equal  and  op- 
posite force,  if  the  weight  is  to  be  in  equilibrium.  From  the 
geometry  of  the  figure,  this  force  may  easily  be  seen  to  be  to 
the  weight  in  the  ratio  of  the  height  of  the  plane  to  its  length. 

14.  Other  Mechanical  Powers. — The  wheel  and  axle,  the 
wedge,  and  the  screw  are  other  arrangements  by  which  me- 
chanical advantage  may  be  obtained,  and  enter  so  frequently 
into  the  construction  of  machinery  and  into  ordinary  me- 
chanical operations  that  they  rank  with  the  lever,  the  pulley, 
and  the  inclined  plane  as  mechanical  powers.  The  mechanical 
advantage  afforded  by  each  of  them  can  be  worked  out  by  con- 


VY£-  It 


20  MECHANICS. 

sidering  them  as  special  cases  of  the  three  mechanical  powers 
which  have  been  already  studied. 

15.  Motion  Due  to  Constant  Force. — When  a  weight  is  not 
sustained   by   an   opposite   force   it   always   falls   toward   the 
earth.     The  rate  at  which  it  falls  evidently  increases,  that  is, 
it  will  fall  faster  and  faster  as  it  nears  the  earth.    The  motion 
of  such  a  body  is  one  of  the  simplest  exhibitions  of  motion 
caused- by  a  force.     It  attracted  the  attention  of  Galileo,  and 
the  study  of  it  by  him  was  the  first  step  taken  in  the  direction 
of  enlarging  the  subject  of  mechanics  to  include  the  study  of 
motions  of  bodies  as  well  as  the  study  of  cases  of  equilibrium. 
In  order  to  give  an  account  of  the  results  reached  by  Galileo, 
certain  preliminary  notions  connected  with  motion  in  general 
must  be  considered. 

16.  Velocity. — When  a  point  moves  along  a  straight  line 
in  such  a  way  that  it  passes  over  equal  distances  in  any  arbi- 
trarily  chosen  'equal    times,    or,   when    a    point   moves    in    a 
straight  line  in  such  a  way  that  the   ratio   of  the   distance 
passed  over  to  the  time  occupied  by  the  point  in  passing  over 
that  distance  is  a  constant,  the  point  is  said  to  have  a  con- 
stant velocity.     When  these  two  conditions,  of  motion   in   a 
straight  line  and  of  a  constant  ratio  between  the  space  and 
the  time,  are  not  fulfilled,  or  when  either  of  them  is  not  ful- 
filled, the  point  is  said  to  have  a  variable  velocity.    The  value 
of  the  variable  velocity,  at  any  point  in  the  line  or  path  over 
which  the  point  moves,  may  be  found  by  supposing  the  moving 
point,  after  it  passes  through  the  point  on  the  line  at  which 
its  velocity  is  desired,  to  move  on  with  a  constant  velocity, 
equal  to  that  which  it  has  as  it  passes  through  the  point  on 
the  line.     This  constant  velocity  will  be  the  velocity  at  the 
point    on    the    line.      In    more    mathematical    language,    this 
velocity  may  be  found  by  taking  the  point  on  the  line  as  the 
origin  from  which  distance  along  the  line  is  to  be  measured, 
forming    the    ratio   between   the    distance    traversed    by    the 
moving  point  along  the  line  and   the  time  taken  by  it  to 
traverse  that  distance,  and  obtaining  the  limit  of  this  ratio  as 
the  time  which  elapses  and  the  space  which  is  traversed  ap- 


MECHANICS.  21 

proach  zero  as  a  limit.  The  numerical  value  of  this  limit  is 
the  magnitude  of  the  desired  velocity,  and  the  direction  of 
the  tangent  to  the  line,  at  the  point  on  it  at  which  the  velocity 
is  desired,  is  the  direction  of  the  desired  velocity.  If  we 
represent  by  s  the  distance  which  the  moving  point  traverses 
in  the  case  of  constant  velocity  and  by  t  the  elapsed  time,  the 

velocity  v  is  given  by  the  formula  v  =  £ . 

The  unit  of  velocity  is  the  velocity  of  a  point  moving  with 
constant  velocity  which  traverses  1  centimetre  in  1  second. 

17.  Acceleration. — When  a  point  moves  with  a  variable 
velocity  it  is  said  to  have  acceleration.     If  the  point  moves  in 
a  straight  line  and  in  such,  a  way  that  its  velocity  changes  by 
equal  amounts  in  any  arbitrarily  chosen  equal  times,  its  ac- 
celeration is  said  to  be  constant.     When  these  two  conditions 
are  not  fulfilled,  or  when  either  of  them  is  not  fulfilled,  the 
acceleration  of  the  point  is  variable.     In  the  case  of  constant 
acceleration,    its   numerical   value   is   found   by    dividing   the 
change    in   velocity    v — v',    which    occurs    in    any    arbitrarily 
chosen    time    t,   by   that   time,    the    quotient   obtained    being 
the  desired  acceleration.     If  we  represent  acceleration  by  a, 

the  formula  by  which  it  is  defined  and  measured  is  «  =  v~v  . 

According  as  the  change  in  velocity  is  an  increase  or  a  de- 
crease, the  acceleration  is  positive  or  negative. 

18.  Effect  of  a  Constant  Force. — Heavy   bodies   evidently 
fall    in    straight   lines   and    with    increasing   velocity.      Their 
motion   is   therefore   accelerated.      The  problem   taken   up   by 
Galileo  was  the  determination  of  the  law  according  to  which 
the  velocity  of  falling  bodies  changes.    It  was  out  of  the  ques- 
tion, with  his  appliances,  to  investigate  this  question  by  a 
direct  measurement  of  the  velocities  of  the  falling  bodies  at 
different  instants.     It  was  therefore  necessary  to   attack   it 
indirectly. 

After  making  an  erroneous  hypothesis  respecting  the  way 
in  which  the  velocity  changes  and  rejecting  it  for  what  seemed 
to  him  good  reasons,  Galileo  assumed  that  the  velocities  are 


22 


MECHANICS. 


proportional  to  the  times  during  which  the  body  has  been 
falling.    He  then  reasoned  as  follows: 

Let  us  suppose  that  a  falling  body  has  a  constant  accele 
ation.     When  it  is  first  released  it  starts  from  rest  and  i 
initial  velocity  (v'  in  the  formula)   is  zero.     After  the  lapse 
of   *    seconds   its   velocity    is   v.     In   the    t   seconds    it   will 
traverse  a  distance  s  which  is  equal  to  the  distance  which  it 
would  traverse  in  t  seconds  if  it  were  moving  with  a  con- 
stant velocity  equal  to  the  average  of  the  velocities  with  which 
it  actually  moves.    On  the  hypothesis  that  its  acceleration  is 
constant  and  therefore  that  its  velocity  has  been  increasing 
at  a  uniform  rate,  this  average  velocity  is  |,    and   the   distance 

vt 
which   it   will   traverse    in   the    t  seconds   is   therefore   s   =  _• 

Since  v  =  at,  we  have  also  s  —  lot2. 

That  this  conclusion  is  correct  may  be  seen  from  the  fol- 
lowing considerations:  Let  us  suppose  that  the  change  in 
velocity  takes  place  not  uniformly,  but  by  small  equal  incre- 
ments occurring  at  the  ends  of  small  equal  intervals  of  time. 
Construct  a  diagram  by  marking  off  along  a  horizontal  line 
equal  distances  representing  equal  increments  of  time,  and  by 
drawing,  from  the  points  thus  determined,  vertical  lines  repre- 
senting the  velocities  which  the  point  will  assume  at  the  cor- 
responding instants.  On  the  lines  thus  determined  construct 
parallelograms.  The  area  of  each  of  these  parallelograms 
will  be  numerically  equal  to  the  product  of  a  time  interval 
and  of  the  velocity  which  the  point  possesses  during  that 
interval,  and  will  therefore  be  the  distance  which  is  traversed 
by  the  moving  point  during  that  interval.  The  sum  of  all  these 
areas  is  therefore  equal  to  the  entire  distance  traversed  by  the 
moving  point  in  the  time  t  under  the  given  conditions.  As  the 
time  intervals  are  taken  smaller  and  smaller  and  the  changes 
in  velocity  become  correspondingly  smaller  and  smaller,  the 
motion  of  the  point  becomes  more  and  more  nearly  that  of  a 
point  moving  with  constant  acceleration  and  the  area  becomes 
more  and  more  nearly  equal  to  the  area  of  the  triangle  whose 


MKCHANICS  23 

base  is  the  line  which  is  numerically  equal  to  t  and  whose 
altitude  is  the  velocity  of  the  point  at  the  end  of  the  time  t. 
In  the  limit,  as  the  time  intervals  vanish,  the  area  which  repre- 
sents numerically  the  distance  traversed  by  the  moving  point 
becomes  the  area  of  the  triangle,  and  since  this  area  is  equal 
to  half  the  base  multiplied  by  the  altitude  of  the  triangle,  we 
reach  the  formula  for  the  distance  traversed  which  has  already 
been  given. 

This  relation  between  the  distance  traversed  by  the  fall- 
ing body  and  the  elapsed  time  may  be  tested  by  experiment, 
and  according  as  the  results  of  experiment  exhibit  this  rela- 
tion or  not,  the  hypothesis  from  which  it  was  deduced  is 
confirmed  or  disproved. 

Galileo's  experiment  consisted  in  allowing  a  smooth  brass 
ball  to  roll  down  an  inclined  plane,  and  in  determining  the 
times  taken  by  the  ball  to  cover  different  distances.  He 
measured  the  time  by  means  of  a  large  vessel,  filled  with 
water,  in  the  bottom  of  which  was  a  small  opening.  This 
opening  was  stopped  with  the  finger  until  the  ball  was  re- 
leased. At  the  moment  the  ball  was  released,  the  finger  was 
removed,  and  the  water  allowed  to  now  out  into  a  small  cup 
until  the  ball  passed  a  marked  point  on  the  plane,  when  the 
opening  was  closed  again.  The  water  that  flowed  into  the 
cup  was  weighed,  and  the  time  which  had  elapsed  Avas  taken 
proportional  to  its  weight.  By  making  a  large  number  of 
experiments  of  this  kind,  to  eliminate  experimental  errors  in 
the  final  averages  taken,  Galileo  was  able  to  show  that  the 
distances  passed  over  by  the  ball  after  starting  from  rest 
were  proportional  to  the  squares  of  the  times  during  which 
it  was  moving. 

In  this  way  it  was  shown  that  the  acceleration  given  to 
falling  bodies  by  their  own  weights  acting  as  forces  is  con- 
stant. 

Galileo  also  showed  that  the  acceleration  of  all  falling 
bodies  which  are  heavy  enough  to  move  through  the  air, 
without  their  motion  being  seriously  affected  by  the  resist- 
ance of  the  air,  is  the  same.  He  did  this  by  allowing  differ- 


24  MECHANICS. 

ent  bodies  to  fall  from  the  top  of  the  Leaning  Tower  of  Pisa, 
releasing  them  at  the  same  instant  and  observing  that  they 
reached  the  ground  together.  This  experimental  result  he 
illustrated  by  the  following  argument:  If  two  exactly  sim- 
ilar bodies,  like  two  coins,  are  released  at  the  same  instant, 
they  will  fall  to  the  ground  in  the  same  time,  and  it  does  not 
seem  reasonable  to  suppose  that  their  motions  will  be  in  any 
way  different  if  they  are  joined  together  so  as  to  form  one 
body. 

The  laws  of  falling  bodies  are  well  illustrated  by  the  use 
of  an  apparatus  called  the  Attwood's  machine.  This  con- 
sists essentially  of  an  easily  running  pulley,  over  which  "is 
passed  a  cord  sustaining  two  equal  weights.  This  system, 
which  is  in  equilibrium,  is  set  in  motion  by  a  small  over- 
weight hung  on  one  end  of  the  cord.  By  suitable  arrange- 
ments the  movements  of  this  system  may  be  examined.  We 
may  show  that  the  velocity  of  the  system,  if  it  starts  from 
rest,  increases  proportionally  to  the  time,  and  that  the  dis- 
tance traversejUby^the  system  is  proportional  to  the  square 
of  the  time. 

From  the  two  formulae  v=at  and  s=$at2,  by  eliminating 
t,  we  obtain  the  relation  v2=2as.  This  relation  is  not  only 
useful  in  solving*  problems  relating  to  falling  bodies,  but  will 
enable  us  in  the  future  to  draw  a  most  important  conclusion. 

19.  Motion  on  an  Inclined  Plane. — When  a  body  moves 
down  an  inclined  plane,  the  force  which  moves  it  is  the  com- 
ponent of  its  weight  which  is  parallel  to  the  plane.  We  as- 
sume, as  Galileo  did,  that  its  acceleration  on  the  plane  will  be 
to  its  acceleration  if  it  is  falling  freely  as  this  component  is 
to  its  whole  weight.  If  we  represent  by  p  the  height  of  the 
plane,  by  s  the  length  of  the  plane,  by  a  its  acceleration  on  the 
plane  and  by  g  its  acceleration  when  falling  freely,  we  have 
the  relation  as=gp>.-  Starting  with  this  relation  we  may  prove 
the  following  theorem:  The  times  of  descent  down  inclined 
planes  of  the  same  height  are  proportional  to  their  lengths. 
For  we  have  s=lat*  and  hence  sz=^gpti,  from  which,  since  g 


MECHANICS.  25 

is  a  constant  and  p  is  the  same  for  all  planes  of  the  same 
height,  we  have  t  proportional  to  s. 

*r\Ve  may  also  prove  the  following  theorem:  The  times  of 
descent  are  equal  down  inclined  planes  whose  lengths  and 
inclinations  are  determined  by  the  chords  of  a  vertical  circle 
drawn  from  its  lowest  point.  That  is,  the  time  of  descent 
down  any  chord  of  a  vertical  circle  drawn  from  a  point  on 
the  circle  to  its  lowest  point  is  the  same  as  the  time  of  free 
fall  through  the  diameter  of  the  circle.  For,  from  the  formula 
s'^lgpt2,  which  was  obtained  in  the  last  paragraph,  the  time  ^/-L.^  / 

t  is   the  same  for  all   inclined   planes  for  which   1  is  the  same, 

P 

From    similar    triangles    in     the    circle    we    have    ^  =  £,    or 

s         d 

d  —  —  ,  and   hence  d  =  IgP.       The  time   t   is  thus  the  time  of 

P 

free  fall  through  the  diameter  of  the  circle,  and  is  the  same 
for  all  the  planes  which  conform  to  the  given  conditions. 

-  'We  may  also  prove  the  following  theorem:  The  velocities 
acquired  in  the  descent  of  a  body  down  different  inclined 
planes  are  equal  if  the  heights  of  the  planes  are  equal.  For, 
from  the  formula  v*=2as  and  the  relation  as=yp,  we  obtain 
the  formula  v2=2gp,  which  shows  that  the  acquired  velocity 
depends  only  on  the  height  of  the  plane  and  not  on  its  length, 
and  is  therefore  the  same  for  all  planes  of  equal  height.  This 
last  relation  holds  even  if  the  path  of  the  moving  body  is 
curved,  as  for  example,  if  it  is  the  arc  of  a  circle;  for,  if  not, 
the  velocity  acquired  will  either  be  greater  or  less  than  the 
velocity  with  which  the  body  must  start  up  an  inclined  plane 
in  order  to  bring  it  to  the  level  from  which  it  started  to  fall. 
If  the  velocity  thus  acquired  is  greater,  the  moving  body  under 
the  action  of  gravity  alone  will  rise  to  a  higher  level  than  that 
from  which  it  starts,  and  this  conclusion  i*  so  contrary  to  our 
experience  that  we  consider  the  hypothesis  upon  which  it  is 
based  to  be  erroneous.  If  the  velocity  acquired  is  less,  it  is 
only  necessary  to  reverse  all  the  motions  in  order  to  reach  the 
same  erroneous  conclusion  from  this  supposition.  We  con- 


2(1  MECHANICS. 

elude  therefore,  that  when  a  body  falls  through  any  path,  the 
velocity  which  it  acquires  in  falling  depends  only  on  the  per- 
pendicular distance  through  which  it  falls  and  not  on  the 
length  or  shape  of  the  path. 

^20.  Composition  of  Velocities  and  Accelerations. — Galileo 
perceived  or  assumed  that  when  a  body  is  in  motion  and  is 
then  acted  on  by  a  force,  the  motion  which  it  will  subsequently 
have  will  be  obtained  by  superposing  on  its  original  motion 
the  motion  given  to  it  by  the  force.  That  is,  the  motion  of  the 
body  is  conceived  of  as  consisting  of  two  motions,  which  exist 
in  the  body  independently  of  each  other. 

Let  us  suppose  that  a  point  is  at  one  instant  in  a  certain 
position  and  at  a  later  instant  in  another  position.  The 
straight  line  which  joins  the  first  position  to  the  second  is 
called  the  displacement  of  the  point.  This  displacement 
manifestly  has  a  determinate  direction  as  well  as  a  magni- 
tude, and  it  has  no  other  characteristics.  It  is  fully  deter- 
mined when  the  direction  and  magnitude  are  given.  A 
quantity  which  conforms  to  these  conditions  is  called  a 
vector. 

Tf  a  point  undergoes  two  successive  displacements,  the 
final  displacement  which  is  the  result  of  the  two,  is-  obtained 
by  drawing  a  line  from  the  position  first  occupied  by  the 
displaced  point  to  the  position  last  occupied  by  it.  This 
line  is  called  the  resultant  displacement.  It  is  manifestly 
the  diagonal  of  the  parallelogram  of  which  the  two  displace- 
ments of  the  point  are  sides.  What  is  true  of  displacements 
is  also  true  of  any  similar  vectors.  The  resultant  of  two 
or  more  vectors  may  be  found,,  or  a  vector  may  be  resolved 
into  components,  by  the  rules  already  given  for  compounding 
and  resolving  forces. 

Velocities,  measured  by  displacements  occurring  in  equal 
intervals  of  time,,  are  vectors,  and  accelerations,  measured 
by  changes  in  velocity  occurring  in  equal  intervals  of  time, 
are  also  vectors  of  another  sort.  Velocities  and  accelera- 
tions may  therefore  be  compounded  and  resolved  by  the 


MECHANICS.  27 

general  rules  which  hold  for  the  composition  and  resolution 
of  vectors. 

21.  Projectiles. — When  a  body  is  thrown  off  in  any  direction 
as  a  projectile,  its  motion  may  be  considered  as  made  up  of  two 
independent  motions,  the  motion  originally  given  it,  which  is 
in  the  original  direction  of  projection,  and  the  motion  im- 
parted to  it  by  the  action  of  its  own  weight  upon  it,  which  is 
directed  vertically  downward. 

We  shall  first  consider  the  simple  case  of  horizontal  pro- 
jection. Suppose  a  body  to  be  projected  horizontally,  from  a 
point  taken  as  origin,  along  the  ar-axis  with  the  velocity  u. 
At  the  end  of  the  time  t  its  displacement  in  the  direction  of 
the  a?-axis  will  be  x=ut.  From  the  instant  of  projection  it 
begins  to  fall  toward  the  earth,  and  if  we  consider  the  t/-axis 
to  be  directed  vertically  downward,  the  distance  which  it  will 
fall  in  the  time  t  in  the  direction  of  the  t/-axis  is  given  by 
y=.\gF.  On  the  supposition  which  we  have  made,  that  the  two 
motions  can  exist  together  in  the  body  without  mutual  inter- 
ference, we  may  obtain  simultaneous  values  of  x  and  y  by 
eliminating  t  between  these  equations.  The  equation  obtained 

is  y  =  jj-^  x2-       This    is    the    equation   of    a.    parabola   with    its 

vertex  at  the  origin  and  represents  the  path  of  the  projectile. 

Since  any  projectile  thrown  obliquely  upwards  will  reach 
a  highest  point  of  its  path,  and  will  at  that  point  be  moving 
'horizontally,  with  a  velocity  equal  to  the  horizontal  com- 
ponent of  the  original  velocity  of  projection,  the  path  which 
it  will  subsequently  traverse  will  be  a  parabola  similar  to 
the  one  already  determined.  Because  of  the  symmetry  of 
the  conditions,  the  path  of  the  projectile  ^before  it  reaches 
its  highest  point  will  be  the  other  branch  of  the  same  para- 
bola. 

To  investigate  the  movement  of  a  projectile  in  this  more 
general  case,  we  suppose  it  to  be  projected  from  the  origin     <-»    v 
obliquely  upwards  with  a  velocity   V,  whose  horizontal  com-  *• 

ponent  along  the  horizontal   a?-axis   is  u  and  whose  vertical 
component  along  the  i/-axis,  directed  upward,  is  v.     Then  in 


28  MKCHANICS. 

the  time  t  the  distance  traversed  along  the  a?-axis  is  x=ut. 
The  distance  which  would  be  traversed  along  the  i/-axis  in  the 
same  time,  if  it  were  not  for  the  change  of  velocity  in  that 
direction  due  to  the  weight  of  the  body,  is  vt.  Owing  to  the 
action  of  the  weight  of  the  body,  the  distance  actually 
traversed  in  the  direction  of  the  y-axis  is  less  than  this,  by 
the  distance  through  which  the  body  will  fall  in  the  time  t. 
It  is  therefore  given  by  y=vt—\gt1.  Eliminating  t  between 
these  equations  we  obtain  the  equation  y  =  -  x  —  ^  *2  as  the 

equation  of  the  path  of  the  body.  It  is  the  equation  of  a 
parabola  which  passes  through  the  origin. 

The  time  of  flight  is  the  time  taken  by  the  projectile,  after 
leaving  the  origin,  to  fall  to  the  same  level  again.  It  may  be 
determined  by  setting  y=Q  and  determining  the  value  of  t 

62n 

obtained  on  this  condition.    We  obtain  t  = 

9 

The  range  of  the  projectile  is  the  distance  between  the 
origin  and  the  point  at  which  the  projectile  again  meets  the 
#-axis.  It  is  therefore  obtained  by  setting  y=Q,  and  determin- 
ing the  value  of  x  obtained  on  this  condition.  The  range  thus 

obtained  is  xf  =  —.  and  is   the  same,  whatever   the  values  of 

9 

u  and  v  may  be,  when  they  are  such  that  their  product  is  con- 
stant. For  a  given  value  of  V,  the  initial  velocity,  there  are 
two  directions  of  projection  for  which  the  product  uv  has  the 
same  value.  These  two  directions  are  equally  inclined  to  the 
line  which  bisects  the  right  angle  between  the  two  axes. 

For  a  given  value  of  V  the  inclination  of  projection  for 
which  the  greatest  range  is  obtained  is  the  inclination  of  this 
bisector.  For,  if  the  angle  of  inclination  of  V  with  the  tf-axis 
is  o,  we  have  «  ==  V  cos  a,  v  =  V  sin  a,  and  2uv  =  2  V2  sin  a  cos  a 
=  V*sin  2a ;  and  this  is  a  maximum  for  a  =  45. 

22.  The  Pendulum. — An  arrangement  which,  for  practical 
reasons,  attracted  special  attention  in  the  early  days  of  the 
study  of  mechanics  is  the  pendulum.  In  the  form  in  which  it 
was  first  studied,  the  pendulum  consists  of  a  small  heavy  body, 


MECHANICS.  29 

which  may  be  considered  a  heavy  point,  swinging  at  the  end 
of  a  long  light  thread  firmly  fastened  at  the  top.  Galileo 
concluded  from  observations  that  the  time  required  for  a 
pendulum  to  execute  one  swing  is  the  same,  whatever  the 
extent  of  the  swing  may  be.  This  conclusion  is  not  strictly 
correct.  Such  observations  as  Galileo  could  make  were  not 
sufficiently  accurate  to  detect  its  falsity,  but  the  fact  is  that 
the  time  of  swing  of  a  pendulum  is  greater  when  the  extent  of 
the  swing  is  greater.  So  long  as  the  extent  of  the  swing  is 
kept  within  certain  limits,  so  that  the  largest  angle  made  by 
the  suspending  thread  with  the  verticul  i.s  not  greater  than  10°, 
the  times  of  swing  are  practically  independent  of  their  extent. 

Galileo  recognized  the  value  of  the  pendulum  as  a  means 
of  measuring  small  intervals  of  time.  Doubtless  it  was  this 
use  to  which  the  pendulum  could  be  put  which  directed 
special  attention  to  it. 

Huygens  applied  the  pendulum  to  the  regulation  of 
clocks,  and  constructed  clocks  which  are  in  all  essential  par- 
ticulars the  same  as  those  of  the  present  day.  He  was  led 
by  his  use  of  the  pendulum  to  investigate  its  properties. 

Huygens  found  that  the  property  of  isochronous  oscilla- 
tion which  is  possessed  approximately  by  the  ordinary  pen- 
dulum swinging  in  a  circular  arc,  is  possessed  exactly  by  a 
pendulum  which  is  so  adjusted  as  to  swing  in  a  cycloidal  arc. 
For  this  reason  the  cycloid  is  called  the  tautochrone.  A 
body  let  fall  from  any  point  on  the  arc  of  a  cycloid  will 
reach  the  lowest  point  of  the  arc  in  the  same  time. 

The  problem  presented  by  the  circular  or  simple  pendu- 
lum is  to  find  the  way  in  which  the  time  of  swing  or  of 
oscillation  depends  upon  the  length  of  the  pendulum. 

It  will  be  shown  hereafter  that  the  time  of  oscillation  of  a 

pendulum  of  length  r  is  given  by  the  formula  t  =  IT   P*.     Since, 

+Jff 
for  any  one  place  on  the  earth,  the  value  of  g  is  constant,  the 

times  of  oscillation  of  pendulums  of  different  lengths  are 
proportional  to  the  square  roots  of  the  lengths.  At  different 
points  on  the  earth's  surface,  where  the  values  of  g  are  differ- 


3,)  MECHANICS. 

ent  the  times  of  oscillation  of  the  same  pendulum  are  inversely 
as  the  square  roots  of  the  value  of  g.  Such  a  difference  in  the 
time  of  oscillation  of  a  pendulum,  due  to  the  different  values 
of  g  at  different  places,  was  first  observed  by  Richer  when  lie 
transported  a  carefully  rated  clock  from  Paris  to  Cayenne 
(1071-1673). 

23.  The  Physical  Pendulum. — Real  pendulums,  especially 
those  used  in  the  regulation  of  clocks,  are  not  made  to  con- 
form to  the  definition  of  a  pendulum  which  has  been  given. 
A  real  pendulum  is  a  body  of  considerable  mass,  which  is  often 
irregularly  shaped.  If  we  consider  each  part  of  this  pendulum 
as  being  itself  the  bob  of  a-  simple  pendulum,  it  is  plain  that 
the  times  of  oscillation  of  these  different  parts,  if  they  were 
free  from  one  another,  would  be  very  different.  As  they  are, 
however,  bound  together  into  one  rigid  body,  some  of  them  are 
forced  to  move  faster  and  others  are  forced  to  move  slower 
than  they  would  if  left  to  themselves.  Manifestly  there  is 
some  one  of  the  parts  which  will  swing  in  the  same  time  as  it 
would  if  left  to  itself,  or  as  if  it  were  the  bob  of  a  simple 
pendulum,  whose  length  is  equal  to  its  distance  from  the  axis 
about  which  the  pendulum  swings.  The  problem  of  finding 
this  length,  which  is  called  the  length  of  the  equivalent  simple 
pendulum,  was  solved  by  Huygens.  He  employed,  as  the  basis 
of  his  solution,  the  principle  that  the  centre  of  gravity  of  a 
system  of  bodies  which  are  left  to  themselves  can  never  rise  to 
a  higher  level  than  that  from  which  it  started.  This  principle 
was  questioned  by  Huygens'  contemporaries,  and  its  applica- 
tion to  the  solution  of  the  pendulum  problem  anticipates  cer- 
tain conceptions  which  may  better  be  brought  out  in  another 
connection.  We  shall  therefore  postpone  the  discussion  of 
Huygens'  result  until  later. 

24.  Newton's  Laws  of  Motion. — We  have  now  reached  cer- 
tain fundamental  conceptions  respecting  the  motion  of  a  body 
when  subjected  to  the  action  of  a  force.  These  conceptions, 
even  when  not  formally  expressed,  have  been  implied  in  all  our 
previous  discussions.  One  of  these  conceptions  is,  that  motion, 
as  well  as  vest,  is  a  natural  state  of  a  body,  and  that  a  body 


MKCHAXICS.  81 

will  persist  in  its  motion  without  change  except  in  so  far  as  it 
is  acted  on  by  a  force.  This  conception  may  be  called  the 
principle  of  inertia.  The  way  in  which  it  was  reached  by  Gali- 
leo is  of  special  interest,  although  his  process  of  thought 
cannot  be  called  a  proof  of  the  principle.  Galileo  conceived  a 
body  to  move  down  an  inclined  plane  and  at  the  bottom  of 
the  plane,  by  a  change  in  direction  which  did  not  involve  a 
change  in  the  body's  velocity,  to  begin  an  ascent  on  a  second 
inclined  plane.  In  accordance  with  the  propositions  we  have 
already  examined,  the  body  will  move  along  the  second  in- 
clined plane  over  a  length  which  will  bring  it  to  the  same 
level  as  that  from  which  it  started  to  fall.  The  time  of  ascent 
through  this  length  will  be  proportional  to  the  length.  Xow, 
as  the  inclination  of  the  second  plane  is  made  less  and  less, 
the  time  of  ascent  wrill  become  longer  and  longer  and  the 
retardation  of  the  body,  or  its  change  of  velocity  in  a  unit 
time,  will  be  less  and  less.  In  the  limiting  case,  in  which  the 
inclination  of  the  plane  is  zero,  the  time  of  ascent  becomes 
infinite  and  there  is  no  retardation,  in  this  special  case,  there- 
fore, in  which  the  moving  body  is  not  acted  on  by  any  force, 
its  motion  will  be  uniform  and  indefinitely  continued. 

Another  of  these  fundamental  conceptions  is  that  the  result 
of  the  application  of  a  constant  force  to  a  body  is  the  produc- 
tion of  a  constant  acceleration.  The  constant  force  considered 
by  Galileo  was  the  weight  of  the  body  or  a  component  of  its 
weight.  Galileo  assumed  that  the  accelerations  produced  in 
the  same  body  by  different  components  of  its  weight,  that  is, 
the  accelerations  of  the  body  when  moving  down  planes  of 
different  inclinations,  are  proportional  to  those  components. 
The  fundamental  conceptions  of  force  and  of  acceleration  as 
produced  by  the  action  of  a  force  were  thus  brought  into  rela- 
tion. This  relation,  however,  was  not  stated,  or  even  assumed, 
as  a  general  one  involving  the  action  of  all  kinds  of  forces, 
because  the  forces  considered  by  Galileo  were  all  of  one  kind. 

In  his  Philosophiae  Naturalis  Principia  Mathematica,  pub- 
lished in  1687,  Newton  collected  the  scattered  and  imperfectly 
formulated  principles  of  his  predecessors,  and  adding  to  them 


32  MECHANICS. 

an  additional  principle,  stated  them  in  three  laws  of  motion, 
which  furnish  a  complete  basis  for  the  science  of  mechanics. 
These  laws  are  as  follows: 

Law  I.  Every  body  continues  in  its  state  of  rest  or  of  uni- 
form motion  in  a  straight  line,  except  in  so  far  as  it  is  com- 
pelled by  an  external  force  to  change  that  state. 

Law  11.  Change  of  motion  is  proportional  to  the  external 
force  applied  and  takes  place  in  the  direction  of  the  applied 
force. 

Law  III.  To  every  action  there  is  an  equal  and  contrary 
reaction;  or,  the  mutual  actions  of  two  bodies  are  always 
equal  and  oppositely  directed. 

The  first  law  simply  states  the  principle  of  inertia  as  it  was 
understood  by  Galileo.  It  is  in  a  sense  contained  in  the  second 
law,  for  it  is  an  evident  implication  of  the  second  law  that, 
if  no  external  force  is  applied  to  a  body,  the  motion  of  the 
body  will  not  change. 

The  second  law  may  be  considered  as  describing  the  way 
in  which  the  motion  of  a  given  body  depends  upon  the  action 
of  forces  of  any  kind  upon  it.  There  is  no  limitation,  in  the 
statement  of  the  law,  to  forces  which  arise  in  any  way  from 
the  weight  of  the  body.  The  forces  which  act  and  which  cause 
the  change  of  motion  may  be  due  to  any  cause  whatever.  By 
the  phrase,  change  of  motion,  is  meant  the  change  of  velocity 
of  the  body  upon  which  the  force  acts,  or  perhaps  better,  the 
acceleration  of  the  body. 

From  this  second  law  we  may  deduce  the  parallelogram  of 
forces;  for,  when  two  forces  act  on  the  body  at  once  they 
will  give  to  the  body  accelerations  in  the  directions  of  the 
forces  and  proportional  to  them.  These  accelerations  are 
vectors  and  may  be  added  by  the  parallelogram  law,  and  since 
the  forces  which  produce  them  are  proportional  to  them,  and 
in  their  directions,  the  forces  are  also  vectors  and  may  be 
added  by  the  same  law.  This  so-called  proof  of  the  parallel- 
ogram of  forces  is  not  logically  more  complete  or  convincing 
than  the  direct  experimental  proof  which  has  already  been 


MECHANICS.  33 

| 

given,  for  it  rests  ultimately  on  the  experimental  evidence  for 
the  truth  of  the  second  law. 

That  part  of  the  second  law  which  states  that  the  forces 
which  act  on  a  body  are  proportional  to  the  accelerations 
which  they  produce,  may  be  illustrated  by  the  Attwood's 
machine.  If  we  change  the  proportionality  between  force  and 
acceleration  into  an  equation  by  introducing  a  constant  or 
factor  of  proportion,  we  have  the  relation  F=ma.  The  ex- 
periments with  the  Attwood  machine  show  that  the  factor  of 
proportion  m  is  proportional  to  the  weight  of  the  system. 

The  fact  that  bodies  of  different  weights  fall  toward  the 
ground  with  the  same  acceleration  confirms  this  conclusion. 
For  if,  in  the  equation  F=mg,  which  applies  to  each  of  these 
different  bodies,  the  acceleration  g  has  a  common  value  and  the 
force  F  is  equal  to  the  weight  of  the  body  in  each  case,  the 
factor  of  proportion  m  must  be  proportional  to  the  weight  of 
the  body  in  each  case.  When  we  confine  our  attention  to  bodies 
of  the  same  material  but  of  different  sizes,  we  perceive  that 
the  quantity  of  matter  in  each  of  these  bodies  is  proportional 
to  the  size  or  volume  of  the  body,  and  in  special  cases  we  may 
increase  our  confidence  in  this  conclusion  by  observing  that 
special  effects  produced  by  the  bodies  are  also  proportional  to 
their  volumes.  In  particular,  the  weights  of  these  bodies  are 
proportional  to  their  volumes.  In  any  such  case,  then,  the 
factors  of  proportion  for  the  different  bodies  are  proportional 
to  the  quantities  of  matter  contained  in  them.  The  conclusion, 
however,  would  not  be  warranted  from  such  observations  that 
the  factors  of  proportion,  in  the  case  of  bodies  of  different 
materials,  are  a  measure  of  the  quantities  of  matter  in  the 
bodies. 

To  draw  this  conclusion  we  must  consider  the  third  law, 
by  means  of  which  a  valid  measure  may  be  obtained  of  the 
quantity  of  matter  contained  in  a  body,  so  far  as  it  relates  to 
the  action  of  force  upon  that  body. 

Before  proceeding  to  the  discussion  of  this  point,  another 
feature  of  the  third  law  must  be  considered.  Up  to  the  time 
of  Newton  no  strict  inquiry  had  been  made  as  to  the  source 


34  MECHANICS. 

or  origin  of  force.  The  physical  universe  was  supposed  to 
consist  of  matter  and  of  forces,  of  which  the  origin  was  un- 
known and  did  not  need  to  be  known.  The  subject  of  study 
was  the  motion  of  a  body  under  the  action  of  given  forces. 
Newton  perceived,  and  embodied  his  perception  in  the  third 
law,  that  a  force  which  acts  on  a  body  always  arises  from  the 
action  of  some  other  body,  and  further  that  this  other  body 
is  also  acted  on  by  a  force  arising"  from  the  first  body.  Thus 
all  action  on  a  body  arises  from  an  interaction  between  it  and 
other  bodies,  and  the  elements  which  constitute  this  inter- 
action, that 'is,  the  forces  which  act  on  the  two  bodies,  are 
equal  and  oppositely  directed. 

This  being  the  case,  it  becomes  possible  for  us  to  compare 
bodies  with  each  other  with  respect  to  the  magnitude  of  the 
proportional  factors  which  characterize  them.  For  when  two 
bodies  interact,  the  product  ma  will  be  the  same  for  each,, 
since  the  forces  which  act  on  them  are  equal  in  magnitude. 
If,  then,  any  one  body  is  taken  as  a  standard,  it  may  be  made 
to  interact  with  other  bodies,  and  the  value  of  m  for  each  of 
these  bodies  may  be  determined  in  terms  of  the  m  of  the 
standard  body.  Experiment  shows  that  the  ratio  of  the  two 
values  of  m  obtained  for  two  bodies  will  be  the  same,  whatever 
body  be  taken  as  the  standard.  The  quantity  m  thus  obtained 
for  any  body  is  called  the  mass  of  the  body.  It  may  be  denned 
as  the  quantity  of  matter  of  the  body,  considered  with  respect 
to  the  effect  produced  upon  it  by  the  action  of  a  force. 

25.  Mass. — The  body  whose  mass  is  taken  as  a  standard  or 
unit  of  mass  is  a  certain  piece  of  platinum,  called  the  standard 
kilogramme.  The  standard  pound,  or  the  mass  of  a  certain 
other  standard  piece  of  platinum,  is  also  often  used  as  a  unit 
of  mass  by  the  English-speaking  nations. 

It  may  be  shown  by  experiment  that  equal  masses,  what- 
ever the  material  of  which  they  are  composed,  have  equal 
weights,  and  it  is  therefore  most  convenient  to  construct  sets 
of  standard  masses  and  their  multiples  and  submultiples  by 
weighing  them.  In  physical  work,  the  unit  mass  which  is  most 
commonly  employed  is  the  one-thousandth  part  of  the  stan- 


MECHANICS.  35 

dard  kilogramme.  This  mass  is  called  the  gramme.  By  the 
aid  of  the  unit  mass  and  of  the  units  of  length  and  time 
already  defined,  units  may  be  constructed  of  all  other  quan- 
tities which  need  to  be  measured  in  mechanics.  These  units 
are  called  absolute  units. 

Jn  particular,  the  absolute  unit  of  force  in  the  C.  G.  S. 
system  is  that  force  which  will  impart  to  a  gramme  one  unit 
of  acceleration,  that  is,  it  is  that  force  which,  acting  uniformly 
for  one  second,  will  impart  to  a  gramme  a  velocity  of  one 
centimetre  per  second.  This  unit  of  force  is  called  the  dyne. 
The  force  which  will  impart  to  a  pound  in  one  second  a  veloc- 
ity of  one  foot  per  second  is  also  taken  as  a  unit  force.  It  is 
called  the  poundal. 

In  practice,  especially  in  engineering  practice  and  on  a 
large  scale,  forces  are  often  measured  in  terms  of  what  is 
called  the  gravitation  unit.  This  unit  is  the  weight  of  a  stan- 
dard mass.  This  standard  mass  may  be  the  gramme,  but  is 
more  often  the  kilogramme,  the  pound,  or  the  ton.  The 
numerical  value  of  a  force  in  terms  of  the  gravitation  unit  is 
its  ratio  to  the  weight  of  the  standard  mass.  Since  the  weight 
of  a  body  is  different  at  different  places  on  the  earth,  the 
gravitation  unit  is  not  everywhere  the  same.  It  differs,  how- 
ever, so  little  in  different  places,  that  its  variations  may  fre- 
quently be  neglected  in  practice. 

The  name  of  the  gravitation  unit  is  the  name  of  the  mass 
whose  weight  is  taken  as  the  unit.  Thus,  we  speak  of  the 
weight  of  a  kilogramme,  or  simply  of  a  kilogramme,  as  a  unit 
force. 

Since  the  weight  of  a  body  in  absolute  units  is  equal  to 
the  mass  of  the  body  multiplied  by  the  acceleration  which  will 
be  imparted  to  it  by  its  weight,  or  since  Weight=mg,  the 
gravitation  unit,  that  is,  the  weight  of  the  unit  of  mass,  is 
equal  to  g  absolute  units  of  force.  The  value  of  g  at  latitude 
40°  is  abo"t  980,  when  the  centimetre  and  the  second  are  the 
units  of  length  and  time.  The  weight  of  a  gramme  is  there- 
fore about  980  dynes.  When  the  foot  and  second  are  taken  is 
the  units  of  length  and  time  the  value  of  g  is  roughly  equal  to 
32.  The  weight  of  a  pound  is  therefore  equal  to  32  poundals. 


36  MECHANICS. 

26.  Units  of  Work. — The  work  of  a  force  has  been  defined 
as  the  product  of  the  force  and  of  the  component  of  the  dis- 
tance through  which  its  point  of  application  moves  in  the 
direction  of  the  force.  If  the  direction  of  the  force  and  the 
direction  in  which  its  point  of  application  moves  are  inclined 
to  each  other  by  the  angle  a,  the  work  of  the  force  is  expressed 
by  the  formula  Fscosa,  in  which  F  represents  the  force  and 
s  the  distance  through  which  its  point  of  application  moves. 
As  may  easily  be  seen  from  this  formula,  the  work  of  the 
force  may  also  be  measured  by  the  product  of  the  distance 
through  which  the  point  of  application  moves  and  the  com- 
ponent of  the  force  in  the  direction  of  motion.  According 
to  the  value  of  the  angle  a,  the  work  of  the  force  may  hav« 
any  value  between  Fs  and  — Fs.  When  the  force  and  the 
motion  of  its  point  of  application  are  perpendicular  to  each 
other,  the  work  of  the  force  is  zero.  When  the  angle  a  is  less 
than  90°,  so  that  cos  o  is  positive,  the  work  of  the  force  is 
positive.  In  this  case  work  is  said  to  be  done  by  the  force. 
When  the  angle  a  is  greater  than  90°,  so  that  cos  a  is  negative, 
the  work  of  the  fcrce  is  negative.  In  this  case  we  may  say 
either  that  negative  work  is  done  by  the  force  or  that  work  is 
done  against  the  force. 

The  absolute  unit  of  work  is  the  work  done  by  unit  force 
,"when  its  point  of  application  moves  through  unit  distance  in 
the  direction  of  the  force.  The  work  done  by  a  dyne  when  its 
point  of  application  moves  in  its  direction  through  one  centi- 
metre is  the  unit  of  work  in  the  C.  G.  S.  system.  It  is  called 
an  erg. 

In  practice  it  is  often  found  convenient  to  use  as  a  unit  of 
work  the  work  done  by  the  gravitation  unit  of  force  when  its 
point  of  application  moves  vertically  through  a  certain  dis- 
tance chosen  as  unit  distance.  Thus  the  work  done  by  a  kilo- 
gramme falling  vertically  through  one  metre  is  often  taken  as 
a  unit  of  work,  it  is  called  the  kilogramme-metre.  The  foot- 
pound, or  the  work  done  by  a  pound  falling  vertically  through 
one  foot,  is  also  a  unit  of  work  which  is  often  employed  by 
the  English-speaking  nations. 


MECHANICS.  37 

27.  Impulse. — If  a  constant  force  F  acts  upon  a  body  of 
mass  m  for  the  time  t,  it  will  impart  to  it  the  velocity  v. 
Since  the  body  in  this  case  has  a  constant  acceleration  equal 

to*,    the  relation  which  connects  these  quantities  is   Ft  =  mv. 

The  product  Ft  is  called  the  impulse.  It  is  of  no  special  im- 
portance when  the  force  which  acts  on  the  body  is  constant,  or 
when  the  force  acts  according  to  a  known  law.  But  it  is 
sometimes  of  considerable  importance  when  the  force  acts  ir- 
regularly or  according  to  no  known  law,  and  especially  when 
it  acts  for  a  very  short  time.  In  such  cases  it  may  not  be 
possible  to  determine  the  force  which  is  acting  at  any  instant, 
but  the  impulse  may  be  determined  by  observing  the  change 
in  velocity  of  the  body  to  which  the  force  is  applied,  and  thus 
an  average  value  of  the  force  during  the  time  during  which  it 
acts  may  be  obtained. 

28.  Momentum. — The   product   of  the   m>i«s   of    a   body   and 
its  velocity  is  called  its  momentum.     The  formula  of  the  last 
section  states  that  the  impulse  applied  to  a  body  is  equal  to 
the  change  which   it  causes   in  the   momentum   of   the   body. 
When  a  constant  force  acts  on  a  body,  as,  for  example,  when  a 
body  falls  toward  the  ground  from  rest,  its  momentum  ac- 
quired is  proportional  to  the  time  during  which  the  force  acts. 

When  two  bodies  act  on  each  other  with  equal  and  opposite 
forces,  according  to  the  third  law,  the  momenta  which  they 
acquire  in  the  same  time  are  eqiial  and  oppositely  directed. 

The  momentum  of  a  body  was  called  by  Descartes  and  by 
Newton  the  quantity  of  motion  of  the  body.  It  measures  the 
quantity  of  motion  in  this  sense,  that  if  bodies  having  differ- 
ent momenta  move  in  directions  opposite  to  equal  constant 
forces  which  act  on  them,  the  times  which  elapse  before  the 
bodies  come  to  rest  are  proportional  to  the  momenta. 

29.  Kinetic   Energy. — When   a   constant   force   acts   on   a 
body~  and  moves  it  from  rest  through  any  distance,  it  doea 
work  equal  to  the  product  of  the  force  and  the  distance.     In 
this  case,  where  there  are  no  counteracting  forces,  the  force  is 
equal  to  the  product  of  the  mass  of  the  body  and  the  accelera- 


206520 


MECHANICS. 


tion  imparted  to  it.  From  the  last  paragraph  of  §18,  the 
velocity  acquired  by  a  body  moving  over  a  distance  s  with  the 
acceleration  a  is  given  by  the  formula  vz=Zas.  Multiplying  by 
m,  and  writing  ma=F,  we  obtain  the  formula  Fs=$mv2,  which 
expresses  the  relation  between  the  work  done  by  a  force  upon 
a  free  body  and  the  velocity  which  the  body  will  acquire.  The 
quantity  expressed  by  %mv*  is  called  the  kinetic  energy  of  the 
body.  It  is  manifestly  not  a  vector,  that  is,  it  does  not  de- 
pend in  any  way  on  direction,  and  it  has  no  negative  values. 
By  a  slight  extension  of  the  demonstration  by  which  this  re- 
lation has  been  obtained,  we  may  show  that,  when  a  body 
has  a  certain  velocity,  and  therefore  a  certain  kinetic  energy, 
on  entering  upon  the  distance  s,  the  kinetic  energy  which  it 
will  have  as  it  leaves  that  distance  differs  from  that  which  it 
had  on  entering  upon  it,  by  an  amount  equal  to  the  work  done 
upon  the  body  as  it  traverses  that  distance.  If  this  work  is 
positive,  the  kinetic  energy  increases  and  the  work  done  is 
equal  to  the  gain  in  kinetic  energy.  If  the  work  is  negative, 
the  kinetic  energy  diminishes  and  the  work  done  is  equal  to 
the  loss  of  kinetic  energy.  Leibnitz  considered  that  the  kinetic 
energy,  or  rather,  the  vis  viva  of  a  body,  that  is,  the  product 
of  its  mass  and  the  square  of  its  velocity,  measured  its 
quantity  of  motion.  Quantity  of  motion  is  so  unintelligible  a 
phrase,  that  the  controversy  which  arose  between  him  and  the 
followers  of  Descartes  about  its  true  significance  could  never 
be  brought  to  a  satisfactory  conclusion.  All  that  can  be  said 
is  that  just  as  there  is  a  sense  in  which  momentum  may  fairly 
be  called  quantity  of  motion  so  there  is  a  sense  in  which  this 
phrase  may  be  applied  to  vis  viva.  Vis  viva  is  quantity  of 
motion  in  this  sense,  that  if  bodies  having  different  values  of 
vis  viva  move  in  directions  opposite  to  equal  constant  forces 
which  act  on  them,  the  distances  passed  over  by  the  bodies 
before  they  come  to  rest  are  proportional  to  the  quantities  of 
vis  viva  of  the  bodies. 

30.  Work  Done  by  Combinations  of  Forces. — When  two  or 
more  forces  act  on  a  body  at  the  same  time,  and  the  body  is 
displaced  through  a  certain  distance,  the  work  done  by  all  the 


MECHANICS.  39 

forces  is  equal  to  the  work  that  would  be  done,  for  the  same 
displacement,  by  their  resultant.  To  show  this  we  shall  con- 
sider first  the  case  of  two  forces  so  acting  that  the  displace- 
ment of  the  body  lies  in  the  plane  of  the  forces.  In  this  case 
it  is  easy  to  see,  by  inspection  of  a  diagram  in  which  the  forces 
and  their  resultant  are  represented  by  the  parallelogram  construc- 
tion, and  the  displacement  is  drawn  from  the  origin  in  its  proper 
direction,  that  the  sum  of  the  components  of  the  two  forces  in 
the  direction  of  the  displacement  is  equal  to  the  component  of 
their  resultant  in  the  same  direction  ;  and  hence  that  the  work 
done  by  the  resultant  is  equal  to  the  work  done  by  the  given 
forces. 

If  the  displacement  of  the  body  is  not  in  the  same  plane 
as  the  forces,  we  may  resolve  it  into  two  components,  one  of 
which  is  in  the  plane  of  the  forces  and  the  other  perpendicular 
to  that  plane.  This  perpendicular  component  will  not  be  in- 
volved in  the  work  done  by  the  forces,  and  the  work  done  by 
them  will  depend  only  upon  the  component  which  lies  in  their 
plane.  To  this  component  the  demonstration  already  given 
applies,  and  the  proposition  already  stated  holds  for  a  dis- 
placement in  any  direction. 

If  itfore  than  two  forces  act,  we  may  proceed  as  in  the 
above  demonstration  with  two  of  the  forces,  and  repeat  the 
demonstration  for  the  resultant  of  these  forces  and  a  third 
force,  and  so  on  until  all  the  forces  have  been  considered.  In 
this  way  we  may  demonstrate  the  general  theorem  that  the 
work  done  by  any  number  of  forces  is  equal  to  the  work  done 
by  their  resultant. 

Jt  may  be  that  one  of  the  two  forces  which  are  doing  work 
on  the  body  is  so  directed  that  the  work  which  it  does  is  nega- 
tive. In  that  case,  on  carrying  out  the  construction  in  the 
same  way  as  has  already  been  described,  it  may  be  proved  that 
the  work  done  by  the  force  which  does  positive  work  is  equal 
to  the  work  done  by  the  other  force,  taken  as  positive,  added 
to  the  work  done  by  the  resultant  of  the  two  forces.  This 
statement  is  true  in  general  for  combinations  of  more  than 
two  forces. 


40  MECHANICS. 

The  result  of  work  done  on  a  body  by  a  combination  of 
forces  is  a  change  in  its  kinetic  energy.  The  magnitude  of  this 
change  is  equal  to  the  work  done  by  the  resultant  and  is  posi- 
tive or  negative  according  as  the  resultant  does  positive  or 
negative  work!  If  the  body  starts  from  rest  under  the  action 
of  the  given  forces,  it  will  move  in  the  direction  of  their  result- 
ant, and  the  kinetic  energy  of  the  body  will  increase. 

The  result  of  this  discussion  may  also  be  stated  by  saying 
that  the  work  done  by  the  forces  which  do  positive  work  is 
equal  to  the  work  done  by  the  forces  which  do  negative  work 
added  to  the  kinetic  energy  gained  by  the  body.  In  the  special 
case  in  which  the  forces  are  in  equilibrium,  so  that  there  is  no 
resultant,  the  kinetic  energy  of  the  body  does  not  change.  In 
this  case  there  is  no  work  done  on  the  body  by  the  system  of 
forces,  considered  as  a  whole,  and  the  work  done  by  the  forces 
which  do  positive  work  is  numerically  equal  to  the  work  done 
by  those  which  do  negative  work. 

31.  Potential  Energy. — There  are  very  many  cases  in  nature 
in  which  the  forces  which  act  on  a  body,  or  at  least  those 
forces  which  are  considered  in  the  study  of  the  body's  motion, 
depend  only  on  the  position  of  the  body  relative  to  other 
bodies  which  act  on  it.  Such  forces  may,  for  the  present,  be 
called  positional  forces.  In  case  the  bodies  upon  which  they 
depend  remain  fixed,  the  positional  forces  which  act  upon  a 
moving  body  as  it  traverses  any  path,  will  be  the  same  in 
whichever  direction  it  is  moving,  and  will  not  depend  upon 
any  peculiarities  of  its  motion.  The  work  which  is  done  on 
the  body  by  those  forces,  when  it  traverses  any  path,  is  there- 
fore numerically  equal  and  of  opposite  sign  to  the  work  which 
is  done  upon  it  when  it  traverses  the  same  path  in  the  oppo- 
site direction. 

When  a  body  is  displaced  in  such  a  way  that  negative  work 
is  done  upon  it  by  positional  forces,  it  is  said  to  have  acquired 
potential  energy.  The  potential  energy  thus  acquired  is  meas- 
ured by  the  work  done  against  these  positional  forces,  or  by 
the  work  which  they  will  do  if  the  body  is  abandoned  to  their 
action  and  retraces  its  path  in  the  opposite  direction. 


MECHANICS.  41 

Other  forces  often  act  upon  a  body  which  do  not  depend 
solely  on  its  position.  Indeed,  in  actual  mechanical  opera- 
tions, such  forces  always  arise,  though  they  are  often  ignored 
in  formal  problems.  When  they  exist,  it  is  found  that  they 
always  depend  upon  the  motion  of  the  body.  They  may  there- 
fore, for  the  present,  be  called  motional  forces.  A  peculiarity 
which  all  motional  forces  possess  in  common  is  that  they  are 
always  directed  oppositely  to  the  direction  of  the  body's 
motion,  so  that  the  work  which  they  do  is  negative.  Work 
done  against  such  forces  does  not  give  to  the  body  potential 
energy,  for  the  body,  if  left  to  itself,  will  not  be  set  in  motion 
or  be  made  to  retrace  its  path  by  these  forces. 

These  positional  and  motional  forces  are  commonly  called 
conservative  and  non-conservative  forces  respectively.  The 
reason  for  these  names  will  be  seen  at  once  if  we 'consider 
these  forces  with  respect  to  the  way  in  which  they  affect  the 
mechanical  energy  of  the  body  upon  which  they  act.  If  the 
only  forces  which  are  acting  are  positional,  the  work  done  by 
those  forces  which  do  positive  work  is  equal  to  the  sum  of  the 
potential  energy  and  the  kinetic  energy  gained  by  the  body. 
In  the  special  case  in  which  no  positive  work  is  done,  the  sum 
of  the  kinetic  energy  and  the  potential  energy  of  the  body 
remains  constant,  or  is  conserved.  If  some  of  the  forces  which 
act  on  the  body  are  motional,  the  work  done  against  them 
has  no  equivalent  in  potential  energy  gained  by  the  body.  In 
this  case,  therefore,  if  there  are  no  forces  doing  positive  work, 
the  sum  of  the  potential  energy  and  the  kinetic  energy  of  a 
body  does  not  remain  constant,  or  is  not  conserved.  In  all 
cases  in  which  motional  forces  occur,  the  sum  of  the  potential 
energy  and  kinetic  energy  of  the  body  continually  diminishes. 

32.  Uniform  Motion  in  a  Circle. — A  body  which  is  moving 
in  a  circle  in  such  a  way  as  to  pass  over  equal  distances  in 
equal  times  does  not  have  a  constant  velocity,  according  to  the 
definition  of  constant  velocity  in  §  16.  The  numerical  value  of 
its  velocity  is  always  the  same,  but  its  direction  is  continually 
changing.  Therefore,  according  to  the  definition  of  accelera- 
tion in  §  17,  such  a  body  has  an  acceleration,  and  from  the 


42  MKCHA.NIO. 

general  relation  between  force  and  acceleration  stated  in  §  24, 
it  must  be  acted  on  by  a  force.  The  problem  of  determining 
the  way  in  which  this  force  depends  on  the  vek>city  of  the 
body  and  the  radius  of  the  circle  in  which  it  moves  was  first 
solved  by  Huygens.  We  shall  investigate  it  by  determining 
the  acceleration  of  a  point  moving  in  the  circle  as  the  body 
moves.  The  force  which  acts  on  the  body  is  then  equal  to  its 
mass,  multiplied  by  this  acceleration. 

Suppose  the  moving  point  to  pass  over  the  small  distance 
AB  in  the  small  time  t.  We  may  consider  this  small  arc  AB 
equal  to  its  chord,  and  we  may  resolve  it  into  two  components, 
-  *£/.  one  of  which  is  tangent  to  the  circle  at  the  point  A,  the  other, 
AD,  perpendicular  to  the  tangent  at  the  same  point.  If  the 
point  had  no  acceleration,  it  would  move  from  the  point  A 
along  the  tangent  to  the  circle  at  A  and  its  displacement  in 
the  small  time  t  would  be  represented  to  a  first  approximation, 
and  in  the  limit,  as  the  time  t  approaches  zero,  would  be  repre- 
sented exactly,  by  the  tangential  component.  Because  of  its 
acceleration,  however,  it  traverses  in  the  time  t  the  distance 
AD  along  the  radius  of  the  circle  toward  its  centre.  We  may 
consider  that  the  acceleration  of  the  point,  with  which  it 
traverses  this  distance,  is  constant  when  the  time  t  is 'small. 
The  distance  AD=s  is  therefore  .given  by  the  formula  s=$at2. 
The  triangles  ABD  and  ACS  are  similar,  and  hence  we  have, 
from  the  proportion  among  their  sides,  *  =  _£_,  from  which  we 
obtain  s  =  |!.  Substituting  this  value  in  the  other  formula 

for  s,  we  obtain  a  =  Jl_  .      Now  c  is  the  distance  traversed  bv 
Pr  * 

the   moving  point   in   the  time   t,   and    H  is  the  velocity  of  the 

moving  point.  We  therefore  obtain  finally  for  the  acceleration 
of  the  point,  which  is  directed  along  the  radius  toward  the 
centre,  the  value  a  =  ^L  The  force  which  acts  on  the  body 

toward  the  centre  is  given  by  —_.      This  force  may  arise  from 


MKCHANICS.  43 

the  action  of  a  body  placed  at  the  centre,  from  the  tension  in 
a  cord  or  other  similar  body  joining  the  moving  body  with  the 
centre,  from  the  pressure  of  a  circular  wall,  or  in  other  ways. 
As  it  is  always  directed  toward  the  centre,  it  is  called  a  cen- 
tripetal force.  • 

From  the  equality  which  we  have  established  between  a 
force  and  the  product  of  the  mass  upon  which  it  acts  and  the 
acceleration  caused  by  it,  it  is  evident  that  if  we  substitute 
for  this  product  a  force  equal  to  it  in  magnitude,  and  opposite 
in  direction  to  the  acceleration,  we  shall  have  a  system  of  two 
forces  in  equilibrium.  Applying  this  method  to  the  case  under 
consideration,  the  fictitious  force  which  we  thus  introduce  is 
equal  to  the  centripetal  force  and  is  directed  away  from  the 
centre  of  the  circle.  It  is  equal  to,  and  in  the  same  direction 
as,  the  real  force  or  reaction  which  is  exerted  upon  a  body  at 
the  centre  of  the  circle,  when  the  centripetal  force  is  due  to 
the  action  of  such  a  body.  The  observation  of  this  reaction, 
which  is  a  real  force,  has  led  to  a  confusion  between  it  and 
the  fictitious  force  which  is  equal  to  it,  and  it  is  commonly 
believed  that  the  moving  body  is  acted  on  by  a  force  tending 
to  carry  it  away  from  the  centre.  This  supposititious  force  is 
called  the  centrifugal  force.  Strictly  speaking,  there  is  no 
such  force  as  a  centrifugal  force,  but  the  term  is  often  a  con- 
venient one  if  it  is  understood  to  mean  simply  the  fictitious 
force  which  may  be  substituted  for  the  product  of  the  mass 
of  the  body  and  its  acceleration. 

33.  Centre  of  Mass. — The  existence  of  a  particular  point  in 
a  body,  which  can  be  determined  from  the  weights  and  the 
relative  positions  of  the  parts  of  the  body,  and  which  is  called 
the  centre  of  gravity,  has  been  explained  in  §  7.  When  the 
notion  of  mass  was  clearly  distinguished  from  that  of  weight, 
it  became  evident  that  a  point  might  be  defined  which  could 
be  called  the  centre  of  mass  of  the  body.  The  definition  of 
this  point  is  analogous  to  that  of  the  centre  of  gravity. 

We  may  define  the  centre  of  mass  of  two  particles,  as  the 
point  on  the  line  which  joins  them,  which  divides  that  line 
into  segments  inversely  proportional  to  the  masses  at  their 


44  MECHANICS. 

extremities.  The  centre  of  mass  of  three  particles  is  obtained 
by  first  locating  the  centre  of  mass  of  two  of  them,  supposing 
£  a  mass  equal  to  the  sum  of  the  two  masses  placed  at  that 
point,  and  then  determining  the  centre  of  mass  between  this 
imaginary  mass  and  the  third  particle.  The  centre  of  mass  of 
a  larger  number  of  particles  is  obtained  by  an  extension  of 
the  process  just  described. 

\Ve  may  define  the  centre  of  mass  otherwise  by  supposing 
the  body  or  system  of  bodies  to  be  composed  of  particles  of 
equal  mass.  On  this  supposition,  the  centre  of  mass  is  the 
point  whose  distances  from  the  three  coordinate  planes  are 
equal  to  the  average  distances  of  these  equal  masses  from  the 
same  planes.  This  definition  is  equivalent  to  the  one  first 
given.  Analysis  shows  that  the  centre  of  mass  thus  deter- 
mined is  a  definite  point,  which  is  independent  of  the  order  in 
which  the  masses  are  combined,  in  carrying  out  the  first  defini- 
tion, and  of  the  position  of  the  coordinate  planes,  in  carrying 
out  the  second  definition. 

It  is  evident  that  this  definition  applies  equally  well  to 
scattered  particles,  or  to  a  collection  of  particles  constituting 
an  extended  body.  It  is  determined  solely  by  the  masses  and 
their  relative  positions,  and  is  consequently  independent  of 
the  circumstances  in  which  the  masses  are  placed.  In  this 
respect  it  differs  from  the  centre  of  gravity.  For  a  small  body, 
the  weights  of  its  parts  are  parallel  forces,  and  are  propor- 
tional to  the  masses  of  the  parts,  so  that  the  definition  of  the 
centre  of  mass,  when  properly  modified,  is  also  the  definition  of 
the  centre  of  gravity  of  such  a  body.  The  centre  of  mass  and  the 
centre  of  gravity  coincide.  But  for  very  large  bodies,  so  extended 
that  the  weights  of  their  parts  are  not  parallel  forces,  there  will  be 
no  true  centre  of  gravity,  except  in  special  cases.  For  such  bodies, 
the  centre  of  gravity  will  depend  on  the  position  of  the  body. 
Even  in  these  bodies,,  however,  the  centre  of  mass  is  a  definite  point. 

The  centre  of  mass  is  of  importance  because  it  is  possible 
to  describe  the  motion  of  a  body  or  of  a  system  of  bodies,  in 
certain  important  respects,  in  terms  of  the  motion  of  the 


MECHANICS.  45 

centre  of  mass.  The  principal  relations  which  illustrate  this 
statement  are  the  following: 

The  velocity  of  the  centre  of  mass  is  equal  to  the  resultant 
of  the  momenta  of  the  different  masses  of  the  system,  divided 
by  the  sum  of  those  masses.  This  is  equivalent  to  saying  that 
the  momentum  of  a  system  of  masses  in  any  direction  is  equal 
to  the  momentum  in  that  direction  of  a  mass,  equal  to  the 
sum  of  the  masses  of  the  system,  moving  with  the  velocity  of 
the  centre  of  mass. 

The  acceleration  of  the  centre  of  mass  is  equal  to  the  re- 
sultant of  the  forces  which  act  on  the  masses  of  the  system, 
divided  by  the  sum  of  those  masses.  This  is  equivalent  to 
saying  that  the  acceleration  of  the  centre  of  mass  is  equal  to 
the  acceleration  which  would  be  imparted  to  a  mass,  equal  to 
the  sum  of  the  masses,  by  a  single  force,  equal  to  the  resultant 
of  the  forces  which  act  on  the  system. 

These  laws  have  been  tacitly  assumed  in  all  the  experi- 
ments in  which  we  have  used  moving  weights  to  illustrate  the 
simple  mechanical  laws.  We  have  treated  the  weight  of  the 
body  in  each  case  as  if  it  were  a  single  force,  and  have  treated 
the  body  as  a  mass  situated  at  a  point.  The  real  forces  were 
of  course  the  weights  of  the  different  parts  of  the  body.  Their 
resultant  was  the  force  which  we  used  as  the  weight  of  the 
body,  and  its  point  of  application,  whose  motion  was  studied. 
was  the  centre  of  mass. 

The  forces  which  act  upon  a  system  of  bodies  arise  either 
from  the  action  of  bodies  outside  the  system,  or  from  the 
interaction  of  bodies  in  the  system.  Such  forces  are  called 
external  and  internal  forces  respectively.  It  is  of  considerable 
importance  to  notice  that  the  motion  of  the  centre  of  mass  is 
not  affected  by  the  action  of  internal  forces.  For,  by  New- 
ton's third  law  of  motion,  these  internal  forces  always  occur 
as  pairs  of  equal  and  opposite  forces.  The  resultant  of  each 
such  pair  of  forces  is  zero,  and  they  therefore  contribute 
nothing  to  the  resultant  force  from  which  the  motion  of  the 
centre  of  mass  is  determined.  If  the  centre  of  mass  is  at  any 
time  at  rest,  it  cannot  be  set  in  motion  by  the  internal  forces 


46  MECHANICS. 

of  the  system,  and  will  therefore  remain  at  rest  unless  exter- 
nal forces  act. 

The  kinetic  energy  of  a  system  of  masses  is  equal  to  the 
kinetic  energy  of  a  mass,  equal  to  the  sum  of  all  the  masses, 
moving  with  the  velocity  of  the  centre  of  mass,  added  to  the 
kinetic  energies  of  the  different  masses  of  the  system,  moving 
with  their  velocities  relative  to  the  centre  of  mass.  That  is, 
if  we  consider  the  centre  of  mass  as  an  origin  from  which  the 
velocities  of  the  parts  of  the  system  may  be  determined,  the 
kinetic  energy  of  the  system  will  be  the  sum  of  the  kinetic 
energies  of  the  parts  determined  from  these  velocities.  If, 
besides  the  motions  of  the  parts  relative  to  the  centre  of  mass, 
the  svstem  as  a  whole  has  a  motion  relative  to  some  external 
point,  the  kinetic  energy  of  the  system  relative  to  that  point 
is  obtained  by  adding  to  the  kinetic  energy  already  obtained 
an  additional  amount  equal  to  the  kinetic  energy  obtained  by 
supposing  all  the  system  concentrated  at  the  centre  of  mass, 
and  moving  with  the  velocity  of  the  centre  of  mass  relative  to 
the  external  point.  It  follows  from  this  statement  that  the 
kinetic  energy  of  a  system  will  never  be  zero  unless  each  part 
of  the  system  is  at  rest.  The  system  as  a  whole  may  have  no 
momentum  even  though  its  parts  are  moving,  provided  its 
centre  of  mass  is  at  rest,  but  it  will  always  have  kinetic  energy 
unless  each  of  its  parts  separately  is  at  rest.  The  reason  for 
this  difference  lies  in  this:  that  momentum  is  a  vector  and 
that  consequently  two  momenta  ^n  opposite  directions  can  be 
added  so  as  to  be  equal  to  no  momentum ;  while  kinetic  energy 
is  not  a  vector,  is  always  a  positive  quantity,  and  BO  when 
two  kinetic  energies  are  added  together  the  result  is  always 
positive. 

34.  Motion  of  Extended  Bodies. — In  describing  the  motion 
of  real  bodies,  it  is  convenient  to  limit  ourselves  to  motion  in 
a  plane.  Motion  in  three  dimensions  is  so  complicated  that 
even  its  fundamental  characteristics  cannot  be  demonstrated 
by  elementary  methods.  In  this  kind  of  motion,  the  parts  of 
the  body  move  in  the  same  plane  or  in  parallel  planes.  The 
formal  object  of  our  study  will  therefore  be  a  plane  figure 


MECHANICS.  '    47 

moving  in  its  own  plane.  At  some  or  all  of  the  points  of  this 
plane  figure  \ve  suppose  masses  placed. 

When  a  plane  figure  moves  so  that  all  its  points  move  in 
similar  and  equal  paths,  its  motion  is  called  a  translation.  A 
translation  is  effected  when  each  point  of  the  body  undergoes 
the  same  displacement. 

When  a  plane  figure  moves  so  that  each  point  of  it.  describes 
an  arc  of  a  circle  around  a  single  point  of  the  plane,  which  is 
the  common  centre  of  all  the  circular  paths  described,  its 
motion  is  called  a  rotation. 

Any  displacement  of  a  plane  figure  in  a  plane  may  be 
obtained  by  the  combination  of  a  translation  and  a  rotacion. 
To  show  this  we  select  a  point,  either  in  the  figure  or  in  an 
extension  of  the  figure,  supposed  to  move  with  it,  as  the  centre 
about  which  rotation  is  to  be  effected.  This  point  is  trans- 
ferred by  a  translation  of  the  figure  from  its  original  position 
to  that  which  it  occupies  in  the  position  given  the  figure  by 
the  displacement.  The  figure  is  then  rotated  around  this 
point  as  centre  into  the  position  which  is  given  it  by  the  dis- 
placement. Since  a  translation  in  any  direction  transfers  any 
straight  line  drawn  in  the  figure  to  a  new  position,  in  which 
that  line  is  parallel  with  its  original  position,  it  is  plain  that 
the  rotation  involved  in  this  operation  will  be  the  same,  that 
is,  will  turn  the  straight  line  through  the  same  angle,  what- 
ever point  in  the  figure  is  taken  as  the  centre  of  rotation. 
The  magnitude  and  the  direction  of  the  translation  will  in  gen- 
eral depend  upon  the  point  chosen  as  the  centre  of  rotation. 

In  particular,  a  point  may  always  be  chosen  as  the  centre 
of  rotation  such  that  no  translation  is  needed  to  effect  its  dis- 
place/nent,  which  can  be  effected  by  a  simple  rotation  around 
that  point  as  centre.  The  construction  by  which  this  point  is 
found  fails  when  the  displacement  can  be  effected  by  a  pure 
translation. 

35.  Rotation. — In  order  to  describe  the  rotation  of  a  body 
and  its  dependence  upon  the  forces  which  act  on  the  body,  it 
is  convenient  at  this  point  to  introduce  a  method  of  describ- 
ing the  motion  of  the  body  in  terms  of  angular  magnitudes. 


48   ,  MECHANICS. 

Consider  a  plane  figure  free  to  rotate  about  a  point.  Draw 
a  line  from  this  point  to  any  definite  point  of  the  figure.  This 
line  merely  marks  a  row  of  points  of  the  figure.  From  the 
same  point  or  centre  of  rotation,  draw  another  line  of  indefi- 
nite length,  which  is  fixed  in  the  plane  in  space  and  serves  as 
an  axis  of  reference  for  the  measurement  of  angles.  The  posi- 
tion of  the  figure  in  the  plane  is  then  determined,  at  any 
instant,  by  the  angle  between  this  axis  and  the  line  first 
drawn  in  the  figure.  When  rotation  takes  place,  the  angle  be- 
tween these  lines  changes,  or  the  figure,  as  determined  by  t.he 
line  drawn  in  it,  rotates  through  a  certain  angle.  This  angle 
is  called  the  angular  displacement.  Each  of  the  points  of  the 
figure  moves  through  an  arc  of  a  circle  whose  radius  is  its 
distance  from  the  centre  of  rotation.  If  the  angle  in  which 
the  angular  displacement  is  given  is  measured  in  radians,  the 
distance  traversed  by  each  point  of  the  figure,  in  the  circular 
arc  which  it  describes,  is  equal  to  its  distance  from  the  centre 
multiplied  by  the  angular  displacement. 

If  the  figure  is  rotating  uniformly,  so  that  its  angular  dis- 
placements in  equal  times  are  equal,  it  has  a  constant  angular 
velocity,  measured  by  the  angular  displacement  which  occurs 
in  a  given  time,  divided  by  that  time.  If  the  angular  displace- 
ment is  not  the  same  for  any  equal  intervals  of  time,  the  angu- 
lar velocity  of  the  figure  is  variable.  Its  value  at  any  instant 
is  the  limit  of  the  ratio  of  the  angular  displacement  which 
occurs  to  the  time  in  which  it  occurs,  as  the  time  and  so  also 
the  angular  displacement  approach  zero.  From  the  relation 
already  described  between  the  distance  actually  traversed  by 
a  point  in  the  figure  and  the  angular  displacement,  it  is  plain 
that  the  velocity  of  any  point  is  equal  to  the  angular  velocity 
of  the  figure  multiplied  by  the  distance  of  that  point  from  the 
centre  of  rotation. 

When  the  angular  velocity  of  the  figure  changes  at  a  uni- 
form rate,  so  that  equal  changes  of  angular  velocity  occur  in 
equal  times,  the  figure  is  said  to  have  a  constant  angular  accel- 
eration. This  constant  angular  acceleration  is  the  ratio  of 
the  change  in  angular  velocity  to  the  time  in  which  that 


MECHANICS.  49 

change  occurs.  If  we  consider  the  motion  of  any  point  of  the 
body,  it  is  plain  that  the  rate  at  which  the  numerical  value  of 
its  velocity  changes  is  equal  to  the  angular  acceleration  mul- 
tiplied by  the  distance  of  that  point  from  the  centre  of  rota- 
tion. 

When  we  consider  a  body  in  rotation  with  constant  angular 
acceleration  a,  the  relation  between  the  time  t  which  has 
elapsed  since  the  body  began  to  rotate  and  the  angular  velocity 
u  acquired  in  that  time,  is  given  by  the  formula  u=at.  From 
the  relation  which  has  been  stated  between  the  rate  of  change 
of  the  numerical  value  of  the  velocity  of  a  point  of  the  figure 
and  the  angular  acceleration,  which  will  make  the  distance 
passed  over  by  any  point  in  the  arc  of  the  circle  in  which  it 
moves  the  same  as  that  which  a  point  would  pass  over  in  a 
straight  line  if  it  were  moving  in  that  line  with  the  ac- 
celeration ra,  we  obtain  fur  that  distance  s  =  r(f>  =  »rat3, 
from  which  we  deduce  the  equation  <f>  —  \a.P  between  the 
angular  displacement  and  the  time  t  in  which  it  has  occurred. 
From  these  equations,  by  eliminating  t,  we  obtain  the  addi- 
tional equation  «2  =  2a0.  These  relations,  expressed  in 
terms  of  the  angular  magnitudes  and  the  elapsed  time,  are 
analogous  to  those  obtained  for  the  motion  of  a  body  moving 
in  a  straight  line  with  a  constant  acceleration. 

36.  Kinetic  Energy  of  a,  Rotating  Body. — A  plane  figure 
rotating  around  a  point,  if  its  points  are  endowed  with  mass, 
is  similar  to  a  body  rotating  around  an  axis  passing  through 
the  centre  of  rotation  and  perpendicular  to  the  plane  of  the 
figure.  If  such  a  body  rotates  with  a  constant  angular 
velocity  u,  it  posesses  kinetic  energy.  The  magnitude  of  its 
kinetic  energy  is  the  sum  of  the  kinetic  energies  of  the  masses 
which  constitute  it.  The  velocity  of  one  of  these  masses,  whose 
distance  from  the  centre  of  rotation  is  r,  is  ?-w,  and  its  kinetic 
energy  is  ^mr^u1.  By  adding  together  the  expressions  such  as 
this  for  the  kinetic  energies  of  the  different  masses  which  make 
up  the  body,  we  obtain  for  the  kinetic  energy  of  the  whole  body 

^—  ~2i  mr*.     The  factor  2  mr*,  which  manifestly  depends  only  on 


50  MECHANICS. 

the  masses  which  make  up  the  body  and  on  their  respective 
distances  from  the  axis  of  rotation,  is  called  the  moment  of 
inertia  of  the  body.  With  respect  to  angular  motions  of  the 
body,  the  moment  of  inertia  plays  the  part  of  the  mass  of  the 
body  with  respect  to  linear  motions. 

37.  Effect  of  Force  on  a  Rotating  Body.  Moment  of  Force. — 
When  a  body  free  to  rotate  is  acted  on  by  a  force,  it  will  in 
general  have  an  angular  acceleration.  The  relation  between 
the  force  and  the  angular  acceleration  which  it  imparts  to  the 
body  may  be  found,  if  we  study  a  simple  case,  from  the  gen- 
eral relation,  established  in  §29,  between  the  work  of  a  force 
and  the  kinetic  energy  produced  by  it.  As  an  example  of  this 
simple  case,  we  may  consider  a  heavy  wheel,  whose  moment 
of  inertia  is  represented  by  /,  mounted  on  a  cylindrical  axle 
whose  radius  is  p,  and  free  to  rotate  about  the  line  which  is 
the  axis  of  this  cylinder.  If  one  end  of  a  flexible  cord  is  at- 
tached to  the  axle,  and  if  the  cord  is  then  wound  several  times 
around  the  axle,  a  weight  hung  on  the  free  end  of  the  cord  will 
be  a  force  so  applied  that  it  will  set  the  wheel  in  rotation. 
Furthermore,  as  the  cord  unwinds,  this  force  will  always  be 
similarly  applied  to  the. axle.  As  the  weight  F  falls  it  does 
work,  the  amount  of  which  is  given  by  Fs.  The  distance  .s 
through  which  the  weight  moves  is  equal  to  the  angular  dis- 
placement 0  of  the  axle  multiplied  by  its  radius  p.  The 
kinetic  energy  acquired  by  the  wheel  during  this  displacement 

is/w_  or,    from    the    equation    of    $3">,     is    Ia<f>.       This   kinetic 

energy  we  may  set  equal  to  the  work  done  by  the  force.  When 
this  is  done,  we  obtain  the  relation  F/)  =  Ia.  The  product  Fp 
is  the  moment  of  force,  as  already  defined  in  §5.  The  relation 
expressed  by  this  equation  may  therefore  be  stated  by  saying, 
that  the  moment  of  force  which  acts  on  a  rotating  body  is  equal 
to  the  moment  of  inertia  of  the  body  multiplied  by  the  angular 
acceleration  imparted  by  the  force.  This  relation  is  analogous 
to  the  fundamental  relation  connecting  force,  mass,  and  accel- 
eration in  linear  motion. 


MECHANICS.  51 

If  several  forces  act  on  a  rotating  body  at  once,  each  of 
them  will  impart  an  angular  acceleration  proportional  to  its 
moment.  These  angular  accelerations  may  be  such  as  either 
to  increase  or  to  decrease  the  angular  velocity  of  the  body. 
Those  which  increase  the  angular  velocity  of  the  body  may  be 
considered  positive,  the  others  negative.  The  algebraic  sum  of 
all  these  angular  accelerations  will  be  the  angular  acceleration 
of  the  body  caused  by  all  the  forces.  If  we  consider  those 
moments  of  force  positive  which  tend  to  turn  the  body  in  one 
sense,  and  those  negative  which  tend  to  turn  it  in  the  op- 
posite sense,  we  obtain  the  relation  S  Fp  =  la,  by  adding  the 
moments  of  force  algebraically.  The  algebraic  sum  S  Fp  of 
the  moments  of  force  is  equivalent  to  a  single  moment  of  force. 

If  the  angular  acceleration  of  the  body  is  zero,  so  that  the 
body  is  in  equilibrium,  we  obtain  2  F/>— .0  as  the  condition  of 
equilibrium.  This  condition  is  the  same  as  that  found  by  ex- 
periment in  §5. 

38.  Couple.  Moment  of  Couple. — A  combination  of  two 
forces  which  are  equal  in  magnitude  and  opposite  in  direction, 
though  not  acting  in  the  same  line,  is  called  a  couple.  When 
a  couple  is  applied  to  a  body  which  is  free  to  rotate  about  an 
axis,  the  moment  of  one  of  these  forces  is  always  greater  than 
that  of  the  other,  since  the  distance  of  one  of  them  from  the 
centre  is  greater  than  that  of  the  other.  The  algebraic  sum 
of  their  moments  is  equal  to  the  product  of  either  one  of  them 
and  the  distance  between  the  lines  in  which  they  act.  Its  sign 
is  positive  or  negative  according  as  the  more  distant  of  the 
two  forces  has  the  positive  or  negative  moment. 

A  couple  has  no  resultant,  and  therefore  no  single  force 
can  be  applied  to  the  body  which  will  counteract  the  effect  of 
a  couple.  A  couple  produces  rotation,  and  the  angular  accel- 
eration imparted  by  it  is  proportional  to  the  moment  of  the 
couple.  Since  the  moment  of  couple  does  not  depend  in  any 
way  on  the  positions  of  the  points  of  application  of  the  forces 
which  constitute  it,  but  only  on  the  magnitude  of  those  forces 
and  the  distance  between  the  lines  in  which  they  act,  the  same 
couple  will  produce  the  same  effect  in  a  body  at  whatever 


52  MECHANICS. 

points  its  forces  are  applied.  And  further,  two  different 
couples  will  produce  the  same  effect  provided  their  moments 
are  the  same. 

A  couple  applied  to  a  body  free  to  move  in  a  plane  will 
produce  rotation  around  the  centre  of  mass.  For,  from  the 
proposition  stated  in  §  33,  the  acceleration  of  the  centre  of 
mass  depends  upon  the  resultant  of  the  forces  which  act  on  the 
body.  Since  the  couple  has  no  resultant,  the  centre  of  mass 
will  have  no  acceleration,  and  the  motion  of  the  body  will 
therefore  be  a  rotation  around  the  centre  of  mass. 

39.  Effect  of  a  Force  on  a  Free  Body. — When  a  force  is  ap- 
plied to  a  free  body,  its  effect  will  be,  in  general,  to  impart 
acceleration  to  the  centre  of  mass  and  also  to  cause  rotation 
around  that  centre,     This  may  be  shown  for  the  plane  figure 
as   follows:      Apply   to   the  centre   of   mass   two   forces   each 
numerically  equal  to  the  given  force,  one  of  them  parallel  to 
the  given  force  and  in  the  same  direction,  the  other  opposite 
to  it.    These  two  forces  being  equal  and  opposite  and  applied 
at  the  same  point,  will  have  no  effect  on  the  motion  of  the 
body.     They  constitute,  along  with  the  given  force,  a  system 
of  three  forces.    According  to  the  proposition  of  §  33  the  appli- 
cation of  the  given  force  to  the  body  will  impart  an  accelera- 
tion to  the  centre  of  mass.     This  acceleration  may  be  con- 
sidered as  arising  from   the  action  of  that   one   of  the  two 
forces  which  is  applied  to  the  centre  of  mass  and  is  parallel  to 
and  in  the  same  direction  as  the  given  force.     The  other  two 
forces  may  be  considered  as  constituting  a  couple  which  will 
produce  rotation  around  the  centre  of  mass. 

40.  The  Physical  Pendulum. — Any  body  which  is  free  to 
rotate  about  a  fixed  horizontal  axis  and  which  swings  back  and 
forth  in  a  vertical  plane  under  the  action  of  its  own  weight,  is 
a  physical  pendulum.     As  was  explained  in  §23,  the  time  of 
oscillation  of  such  -a  pendulum  must  be  the  same  as  that  of  a 
certain  simple  pendulum.     The  problem  before  us  is  to  deter- 
mine, from  the  characteristics  of  the  physical  pendulum,  the 
length  of  the  simple  pendulum  whose  period  of  oscillation  is 
the  same  as  that  of  the  physical  pendulum.    When  the  pendu- 


MECHANICS.  53 

lum  is  not  swinging,  the  line  drawn  from  the  axis  of  sus- 
pension to  its  centre  of  gravity  is  vertical.  When  the  pendu- 
dum  swings,  the  extent  of  its  swing  is  measured  by  the  angle 
<f>  which  this  line  drawn  in  the  pendulum  makes  with  the 
vertical.  The  moment  of  force  which  acts  upon  the  pendulum 
when  its  deviation  is  <j>  is  its  weight  Mg  applied  at  its  centre 
of  gravity  multiplied  by  the  distance  R  sin<£  from  the  axis 
of  suspension  to  the  line  of  direction  of  the  weight.  This  be- 
ing so,  the  angular  acceleration  of  the  pendulum  varies  with 
<f>  and  is  given  at  the  instant  at  which  its  deviation  is  0  by 
the  formula  Mg.Rsin  0=/o.  The  motion  of  the  pendulum 
is  therefore  such  that  its  angular  acceleration  is  proportional 

to  the  ijine  of  its  deviation,  and  the  factor  M9R  \s   tne    factor 

of  proportion. 

Considering  now  the  case  of  the  simple  pendulum,  we  see 
that  the  acceleration  of  the  bob,  tangent  to  its  circular  path, 
is  the  component  of  the  acceleration  g  in  that  direction.  This 
component  is  given  by  g  sin  <f>.  The  acceleration  is  equal  to 
the  angular  acceleration  of  the  pendulum  multiplied  by  its 
length  r.  Introducing  this  value  for  it,  we  obtain  for  the 

angular  acceleration    the   formula  a  =  ?.  sin  <j>.       A     comparison 

of  this  formula  with  the  one  obtained  for  the  physical  pendu- 
lum in  the  preceding  paragraph  shows  that  when  the  deviations 
of  the  two  pendulums  are  the  same,  that  is,  when  the  values  of 
<(>  are.  the  same  in  both  formulae,  the  accelerations  will  be  the 
same  and  therefore  the  motions  will  be  in  every  respect  the 

same,  if  the  factor  of  proportion  ff  is  equal  to  the  corre- 
sponding factor  of  proportion  €.  Setting  these  two  quantities 

T. 

equal,  we  obtain  for  the  length  of  the  simple  pendulum  which 
will  swing  in  the  same  period  as  the  physical  pendulum  the 

expression  r  — .      The  quantity  7  is  the  moment  of  inertia 

MR 


51  MECHANICS. 

of  the  pendulum  about  its  axis  of  suspension.     The  quantity 
AIR  is  called  the  static  moment. 

The  problem  of  finding  the  period  of  oscillation  of  the 
physical  pendulum  is  thus  reduced  to  the  problem  of  finding 
the  period  of  the  equivalent  simple  pendulum.  This  problem 
cannot  be  solved  by  elementary  methods  in  the  general  case, 
when  no  limit  is  placed  on  the  deviation.  We  shall  therefore 
consider  only  the  special  case  in  which  the  deviation  is  always 
so  small  that  the  arc  expressing  it  may  be  substituted  for  its 
sine.  In  this  case,  we  may  state  the  condition  to  which  the 
motion  of  the  pendulum  always  conforms  by  saying  that  its 
angular  acceleration  is  everywhere  proportional  to  its  angular 
displacement.  If  we  multiply  both  sides  of  the  equation 

a  =  ^  rf>,  expressing   this  relation,  by   r,  we  obtain  on   the  left 

r 
the  acceleration  of  the  pendulum  bob  in  its  path,  and  on  the 

right  the  factor  of  proportion  *?  multiplied   by   the  displacement 

s  =  r  <f>  of  the  bob  from  the  centre  of  its  path.     The  acceleration 
of  the  bob  is  therefore  proportional  to  its  displacement  and 

the  factor  of  proportion  is  •?. 

By  the  following  construction  we  may  describe  a  motion 
which  possesses  this  characteristic  relation  between  the  ac- 
celeration and  the  displacement,  and  from  it  we  may  determine 
the  period  of  the  pendulum.  Draw  the  straight  line  LOK  to 
represent  the  path  of  the  pendulum  bob.  Upon  it  as  diameter 
construct  a  circle.  Take  any  point  A  in  this  circle  and  let 
fall  the  perpendicular  AB  upon  the  diameter.  If  the  point  A 
moves  with  the  numerically  constant  velocity  v  around  this 
circle,  we  may  prove  that  the  motion  of  the  point  B,  which  is 
the  point  of  intersection  of  the  diameter  LK,  and  the  perpen- 
dicular let  fall  upon  it  from  the  moving  point  A,  will  be  such 
that  its  acceleration  will  be  proportional  to  its  distance  OB 
from  the  centre  of  the  circle,  or  to  its  displacement.  For,  the 
acceleration  of  the  point  B  is  the  component  along  the  diameter 
of  the  acceleration  of  the  point  A.  The  acceleration  of  the 


MECHANICS.  55 

point  A  is  directed  along  the  radius  OA  toward  the  centre  and 

is  equal  to  *  ,  if  a  is  the   length   of  the  radius.      Its  component 
a 

parallel  with  the  diameter  has  the  same  ratio  to  the  accelera- 
tion of  the  point  A  as  the  displacement  OB=s  has  to  the 

radius  a,   or  is  equal  to  —  .  -.        Comparing   the    motion   of  the 
a   a 

point  B  with  that  of  the  pendulum  bob,  we  see  that  when  the 
displacements  are  the  same,  the  accelerations  will  be  the  same 

at    each    point    when    the    factors    of    proportion  !L  and  ?  are 

equal.  And  also,  in  this  case,  the  period  in  which  the  pendu- 
lum bob  describes  its  path  is  the  same  as  the  period  of  the 
point  B.  Now,  on  the  suppositions  by  means  of  which  the 
motion  of  the  point  B  has  been  described,  we  may  determine 
the  period  T  of  the  point  B.  It  is  evidently  the  time  taken  by 
the  point  A  to  describe  the  circle.  The  velocity  of  the  point  A 
is  the  ratio  of  the  circumference  to  the  period  and  is  given  by 

v=  _?[?.  Substituting  this  value  of  v  in  the  equation  V—  =  ?> 
we  obtain,  for  the  value  of  the  period,  T=  2v  It.  This  is  the 


be 


time  required  by  the  simple  pendulum  to  execute  a  double 
vibration.  The  time  required  to  execute  a  single  vibration, 

which  is  that  commonly  observed,  is  t  —  ir     IL.      It   is   to 

W0 

noticed  that  this  value  of  the  period  is  independent  of  the  dis- 
placement of  the  pendulum,  and  will  therefore  be  the  same 
for  any  displacement  within  the  limits  assumed  at  the  outset 
of  the  discussion. 

The  point  in  the  physical  pendulum  whose  distance,  meas- 
ured along  the  line  drawn  from  the  point  of  suspension 
through  the  centre  of  gravity,  is  equal  to  the  length  of  the 
simple  pendulum  which  will  swing  in  the  same  period,  is  called 
the  centre  of  oscillation  of  the  pendulum.  It  possesses  an- 
other peculiarity,  in  consequence  of  which  it  may  also  be  called 
the  centre  of  percussion.  If  an  impulse  is  applied  horizon- 


56  MECHANICS. 

tally  to  it  when  the  pendulum  is  at  rest,  the  motion  which 
occurs  is  a  pure  rotation  about  the  axis  of  suspension,  or 
brings  no  strain  on  the  axis.  Furthermore,  if  the  pendulum 
is  reversed  and  made  to  swing  about  an  axis  passing  through 
the  centre  of  oscillation,  its  period  will  be  the  same  as  before. 
A  pendulum  so  adjusted  that  its  periods  of  oscillation  about 
two  points  of  suspension,  both  of  which  lie  on  a  line  passing 
through  the  centre  of  gravity,  are  equal  is  called  a  reversible 
pendulum.  The  distance  between  these  points  of  suspension 
is  the  length  of  the  simple  pendulum  which  has  the  same 
period. 

41.  Motion  in  Three  Dimensions. — The  motion  of  a  body  in 
three  dimensions  may  be  described  in  a  way  which  is  in  some 
respects  analogous  to  that  employed  for  the  description  of  the 
motion  of  a  plane  figure.  Any  displacement  of  a  body  may  be 
accomplished  by  a  translation  and  a  rotation  around  a  suit- 
ably chosen  axis.  A  direction  of  translation  may  always  be 
found  such  that  the  axis  around  which  the  necessary  rotation 
takes  place  is  in  the  same  direction.  An  infinitesimal  dis- 
placement of  the  body  may  thus  be  analyzed  into  an  infinites- 
imal translation  and  an  infinitesimal  rotation  around  an  axis 
drawn  in  the  same  direction  as  the  translation.  This  motion 
is  that  of  a  part  of  a  screw,  when  the  screw  is  turned  and  so 
driven  forward,  and  the  motion  of  a  body  at  any  instant  may 
therefore  be  called  a  screw  motion.  As  the  body  moves 
through  a  finite  path,  the  characteristics  of  the  screw  whose 
motion  describes  the  motion  of  the  body,  that  is,  the  pitch  of 
the  screw  and  the  direction  of  its  axis,  change  from  instant 
to  instant.  The  changes,  however,  are  never  discontinuous. 

The  study  of  the  moment  of  inertia  of  a  body  around  an 
axis  passing  through  the  centre  of  mass  shows  that  this 
moment  of  inertia  can  always  be  found  if  we  know  the  mo- 
ments of  inertia  around  three  principal  axes  which  pass 
through  the  centre  of  mass  and  are  perpendicular  to  each 
other.  Around  one  of  these  axes  the  moment  of  inertia  has 
a  maximum  value,  around  one  of  the  others  a  minimum  value. 
When  the  body  is  rotating  around  its  axis  of  greatest  moment 


MECHANICS.  57 

of  inertia,  or  around  its  axis  of  least  moment  of  inertia,  it  is 
in  a  condition  of  kinetic  stability,  that  is,  an  impulse  applied 
to  the  body,  though  it  will  compel  the  body  to  rotate  around 
a  new  axis,  will  not  so  alter  the  motion  of  the  body  as  to  cause 
the  new  axis  to  deviate  more  and  more  from  the  direction  of 
the  original  axis.  If  the  original  rotation  is  around  the  third 
of  the  principal  axes,  its  condition  is  unstable,  that  is,  though 
the  rotation  will  persist  around  that  axis  so  long  as  no  im- 
pulse is  applied  to  the  body,  the  application  of  an  impulse  will 
cause  a  rotation  around  a  new  axis,  whose  direction  will  con- 
tinually deviate  more  and  more  from  that  of  the  .original  axis. 
A  rotation  set  up  aro«nd  any  other  axis  than  one  of  these 
three  will  not  continue  around  that  axis,  even  though  no  im- 
pulse is  applied  to  the  body,  but  the  axis  around  which  rota- 
tion occurs  will  change  its  direction  in  the  body  continually. 

When  a  body  is  in  rotation  about  its  axis  of  greatest 
moment  of  inertia,  the  application  of  a  small  couple,  to  pro- 
duce rotation  about  a  perpendicular  axis,  will  result  in  such 
a  combination  of  motions  that  the  final  effect  is  a  rotation 
about  a  third  axis  perpendicular  to  both  the  others.  This 
statement  is  illustrated  by  the  instrument  called  the  gyro- 
scope or  gyrostat.  Rotation  of  a  body  about  an  axis  of  great- 
est moment  of  inertia  thus  introduces  a  resistance  to  any 
force  which  is  so  applied  as  to  change  the  direction  of  that 
axis.  Thus  if  a  heavy  wheel  is  mounted  inside  a  box  and  is 
kept  in  continual  rotation,  the  box,  though  of  itself  it  does 
not  originate  motion,  nor  resist  a  motion  of  translation,  will 
offer  a  resistance  to  any  force  tending  to  turn  it  around.  Cer- 
tain forces  which  exist  in  nature  have  been  explained  by  sup- 
posing that  the  bodies  which  exhibit  them  contain  portions 
W7hich  are  in  rapid  rotation.  Since  these  rotating  portions, 
even  if  they  exist,  are  such  that  they  can  never  be  perceived, 
they  are  said  to  be  concealed,  and  the  forces  exhibited  by  the 
bodies  are  said  to  be  due  to  concealed  motion. 

'The  angular  velocity  of  a  body  rotating  around  an  axis 
may  be  conceived  of  as  made  up  of  two  angular  velocities 
about  other  axes  passing  through  this  axis  at  the  same  point. 


58  MECHANICS. 

The  magnitudes  of  these  component  angular  velocities,  as  they 
may  be  called,  may  be  found  by  laying  off  on  the  original 
axis,  from  the  origin  or  point  at  which  the  three  axes  inter- 
sect, a  distance  which  is  numerically  equal  to  the  angular 
velocity  about  that  axis,  and  constructing  upon  this  distance 
as  diagonal  the  parallelogram  whose  sides  are  in  the  directions 
of  the  other  axes.  This  statement  may  be  illustrated  by  the 
Foucault  pendulum.  This  pendulum  consists  of  a  heavy  bob 
suspended  by  a  long  cylindrical  wire.  The  upper  end  of  the 
wire  is  held  in  a  clamp  in  such  a  way  that  it  may  swing  with 
equal  freedom  in  any  direction.  Arrangements  are  made  for 
noting  the  plane  in  which  the  pendulum  swings  at  any  in- 
stant. The  pendulum  thus  arranged  is  set  swinging,  as  ex- 
actly as  possible,  in  a  vertical  plane,  and  the  plane  of  its 
swing  is  noted  from  time  to  time.  It  is  found  that  this  plane 
changes  its  apparent  direction  with  respect  to  the  surface  of 
the  earth,  and  that  the  angular  deviation  of  the  plane  of  swing 
from  its  original  position,  in  a  given  time,  is  at  different 
places  proportional  to  the  sine  of  the  latitude  of  the  place. 
We  may  give  a  kinematic  description  of  this  result  as  follows: 
If  we  conceive  such  a  pendulum  set  up  at  the  North  Pole,  the 
plane  of  its  swing  will  remain  fixed  in  space  and  the  earth  will 
turn  around  under  it,  so  that  to  an  observer  examining  the 
pendulum,  its  plane  of  swing  will  change  its  position  relative 
to  the  earth's  surface.  In  one  day  it  will  appear  to  have 
traversed  a  complete  circle.  If  the  same  pendulum  is  set  up 
at  the  .Equator,  and  is  set  swinging  in  the  north  and  south  line, 
the  earth  will  carry  it  around,  and  its  plane  of  swing  will  not 
change  its  position  relative  to  the  earth.  At  any  intermediate 
station  the  motion  of  the  plane  of  swing  may  be  determined 
as  follows:  Draw  a  line  from  the  centre  of  the  earth  to  the 
station  and  another  line  perpendicular  to  this,  in  the  plane 
containing  this  line  and  the  earth's  axis.  We  may  conceive 
the  angular  velocity  of  the  earth  resolved  into  two  angular 
velocities  around  these  two  axes.  The  angular  velocity  around 
the  perpendicular  axis  last  drawn  will  have  no  effect  upon  the 
motion  of  the  pendulum.  The  angular  velocity  about  the  other 


MECHANICS.  "59 

axis,  however,  which  is  equal  to  the  angular  velocity  of  the 
earth  multiplied  by  the  sine  of  the  latitude,  may  be  considered 
as  an  angular  velocity  with  which  the  earth  turns  under  the 
pendulum  around  the  axis  passing  through  the  station.  The 
plane  of  swing  will  therefore  have  an  apparent  angular  velocity 
around  this  axis  which  is  proportional  to  the  sine  of  the  lati- 
tude. The  full  dynamical  discussion  of  the  motion  of  the  Fou- 
cault  pendulum  cannot  be  given  here. 


MECHANICS   OF   LIQUIDS. 


MECHANICS  OF  LIQUIDS. 

42.  The  Problem  of  the  Crown. — The  study  of  the  peculiar 
mechanical  effect*,  exhibited  by  liquids  was  "begun  by  Archi- 
medes. According  to  the  story  told  by  Vitruvius,  a  certain 
quantity  of  gold  had  been  given  by  King  Hiero  to.  a  goldsmith, 
with  the  order  to  use  it  in  constructing  an  elaborate  crown. 
The  crown  when  returned  weighed  as  much  as  the  gold  which 
had  been  supplied,  but  for  some  reason  Hiero  suspected  that 
a  part  of  the  gold  had  been  abstracted,  and  silver  substituted 
for  it.  He  asked  Archimedes  to  devise  a  way  to  determine 
whether  this  was  so  or  not,  without  injuring  the  workmanship 
of  the  crown.  Archimedes  is  said  to  have  discovered  the  way 
in  which  this  might  be  done  by  noticing  the  way  in  which  the 
water  overflowed  from  a  bath  into  which  he  had  entered.  It 
is  plain  that,  if  bodies  of  different  materials  and  of  equal 
weights  do  not  occupy  equal  volumes,  the  one  having  the 
larger  volume  will  cause  more  water  to  overflow  from  a  full 
vessel  in  which  it  is  immersed  than  the  other  one  will,  and 
that  by  determining  the  overflow  caused  by  the  crown  and  by 
equal  weights  of  gold  and  silver,  the  question  asked  of  Archi- 
medes might  be  answered. 

By  reflection  upon  the  question  thus  presented,  Archimedes 
was  led  to  assume  certain  principles  to  which  a  liquid  will 
conform,  and,  by  the  aid  of  these  principles,  to  deduce  certain 
laws  which  govern  the  apparent  loss  of  weight  of  bodies  im- 
mersed in  a  liquid,  and  to  determine  certain  cases  of  equil- 
iibrium  of  floating  bodies.  These  principles,  which  were  made 
by  Archimedes  the  postulates  of  his  theory,  were,  that  when 
two  portions  of  a  liquid  are  similarly  situated  and  are  con- 
tiguous to  each  other,  the  portion  which  is  under  less  pressure 
is  set  in  motion  by  the  portion  which  is  under  greater  pressure; 
and  that  the  pressure  at  a  point  in  the  liquid  is  proportional 
to  the  height  of  the  column  of  liquid  which  stands  vertically 
above  it.  These  principles  are  not  so  fundamental  as  the 


MECHANICS   OF   LIQUIDS.  61 

principle  afterwards  introduced  by  Pascal.  We  shall  accord- 
ingly pass,  without  further  consideration  of  them,  to  a  state- 
ment of  Pascal's  principle  and  an  investigation  of  its  conse- 
quences. 

43.  Pascal's  Principle. — According  to  Pascal  the  character- 
istic peculiarity  of  a  liquid  may  be  expressed  by  saying  that, 
when  a  liquid  is  under  the  action  of  no  forces  except  pressures 
applied  to  its  external  surface,  the  pressure  at  every  point  in 
it  and  in  every  direction  around  each  point  is  the  same.  To 
properly  appreciate  Pascal's  principle,  we  must  clearly  define  a 
pressure  and  understand  how  it  may  be  measured.  To  do  this 
in  the  simplest  possible  way,  let  us  suppose  that  we  have  a 
cylindrical  vessel  closed  above  by  a  tightly  fitting  piston,  that 
can  move  in  the  cylinder,  and  that  the  space  below  the  piston 
is  filled  with  a  liquid.  If  weights  are  put  upon  the  piston, 
their  combined  effect  is  a  resultant  force  applied  to  the  piston 
at  a  point  below  their  centre  of  gravity.  By  a  suitable  adjust- 
ment of  the  weights,  this  centre  of  gravity  may  be  made  to 
coincide  with  the  centre  of  figure  of  the  piston.  The  piston, 
with  the  weights  on  it,  is  in  equilibrium  under  the  action  of 
this  resultant  force  and  of  an  equal  and  opposite  resultant, 
which  arises  from  the  action  upon  the  piston  of  the  different 
parts  of  the  liquid  in  contact  with  it.  Because  of  the  similar- 
ity of  the  parts  of  the  liquid  to  one  another,  the  forces  which 
they  exert  are  thought  of  as  equal  and  uniformly  distributed 
over  the  piston.  If  this  be  so,  the  resultant  force  which  is 
applied  to  unit  area  of  the  piston  is  found  by  dividing  the 
total  force  applied  to  the  piston  by  its  area.  This  resultant 
force  applied  to  unit  area  is  called  the  pressure  of  the  liquid 
on  the  piston.  Obviously  it  is  possible  to  obtain  the  same 
value  for  the  pressure  by  supposing  the  forces  applied  to  the 
piston  to  be  increased  in  number  and  diminished  in  magnitude 
without  limit,  on  the  condition  that  the  sum  of  all  these 
forces  remains  constant  and  taking  the  ratio  between  the  re- 
sultant force  which  acts  on  any  area  and  that  area.  This 
statement  is  important  in  defining  pressure  at  a  point,  which 


52  MECHANICS   OF    LIQUIDS. 

is  the  limit  of  the  ratio  just  stated  as  the  area  approaches 
zero. 

To  conceive  of  a  pressure  in  a  liquid  we  may  suppose  a 
plane  surface  drawn  in  the  liquid  and  the  liquid  removed  on 
"  one  side  of  it.  To  keep  the  surface  in  equilibrium  forces  would 
have  to  be  applied  to  it,  imagining  it  to  be  made  up  of  rigid 
parts.  From  these  forces  and  the  areas  to  which  they  are 
applied,  the  pressure  may  be  measured. 

A  statement  which  is  even  more  fundamental  than  Pascal's 
principle  is  that  the  pressure  on  any  surface  in  a  liquid  is 
always  normal  to  it.  From  this  principle  Pascal's  principle 
may  be  deduced. 

An  application  of  Pascal's  principle  is  made  in  the  hydro- 
static press.  This  consists  essentially  of  two  cylinders  joined 
by  a  connecting  pipe.  The  diameter  of  one  of  these  cylinders 
is  considerably  larger  than  that  of  the  other.  The  cylinders 
and  the  pipe  connecting  them  are  filled  with  water,  and  pis- 
tons are  inserted  which  rest  upon  the  surfaces  of  the  water 
in  the  cylinders.  If  weights  are  placed  upon  the  pistons  they 
wTill  be  balanced  if  they  are  to  each  other  in  the  ratio  of  the 
areas  of  the  pistons  on  which  they  stand.  The  condition  of 
equilibrium  which  has  been  stated  follows  from  Pascal's  prin- 
ciple, for  from  that  principle,  the  pressure  on  unit  area  is  the 
same  for  each  piston,  and  the  total  pressures  on  the  pistons, 
or  the  weights  which  they  will  sustain,  are  therefore  propor- 
tional to  their  areas. 

44.  Pressure  in  a  Liquid  Due  to  Its  Weight. — When  a  liquid 
stands  at  rest  in  a  vessel,  the  pressure  at  any  point  within  it 
is  proportional  to  the  depth  of  that  point  below  the  surface. 
Let  us  imagine  a  circle  of  unit  area  placed  at  the  point  and 
parallel  with  the  surface.  The  cylindrical  column  of  liquid 
which  stands  on  that  area  is  in  equilibrium  under  the  com- 
bined action  of  its  .weight,  of  the  upward  pressure  applied  to 
the  area  on  which  it  stands,  and  of  the  pressures  applied  to  its 
sides.  The  pressures  applied  to  its  sides  are  perpendicular  to 
the  cylindrical  surface  and  exert  equal  and  opposite  forces 
upon  this  cylinder,  so  that  they  do  not  counteract  the  weight 


MECHANICS    OF    LIQUIDS.  63 

of  the  column.  The  weight  is  counteracted  only  by  the  up- 
ward pressure  applied  to  the  base  of  the  cylinder,  and  since 
the  weight  is  proportional  to  the  height  of  the  cylinder,  the 
pressure  is  also  proportional  to  that  height. 

The  free  surface  of  a  liquid  is  a  horizontal  plane.  If  we 
describe  a  cylinder  in  the  liquid  around  a  horizontal  line  as 
axis,  the  pressures  acting  on  its  two  ends  are  equal;  for,  the 
cylinder  is  in  equilibrium  along  its  axis  under  these  pressures 
alone,  since  its  weight  has  no  horizontal  component.  The  two 
points  at  which  the  pressures  are  equal  will,  from  the  previous 
proposition,  be  at  equal  distances  from  the  surface,  and  the 
surface  is  therefore  horizontal. 

If  the  mass  of  liquid  is  so  extended  that  the  weights  of  its 
different  parts  are  not  parallel,  but  converge  toward  the 
centre  of  the  earth,  a  slight  modification  of  the  demonstration 
just  given  will  show  that  the  free  surface  is  part  of  the  surface 
of  a  sphere,  whose  centre  is  the  centre  of  the  earth.  This 
form  of  the  proposition  was  demonstrated  by  Archimedes. 

From  this  proposition  it  may  easily  be  seen  that  a  liquid 
in  two  communicating  vessels  will  stand  at  the  same  level  in 
both. 

45  Archimedes'  Principle. — One  of  Archiinede*'  proposi- 
tions is  of  such  fundamental  importance  that  it  is  usually 
recognized  as  his  especial  contribution  to  hydrostatics,  and 
called  Archimedes'  principle.  We  may  state  this  principle  as 
follows:  When  a  body  is  immersed  in  liquid  it  loses  in  ap- 
parent weight  an  amount  equal  to  tbe  weight  of  the  liquid 
displaced  by  it.  Of  course  the  body  does  not  really  lose  weight, 
but  part  of  its  weight  is  counteracted  by  the  upward  pressure 
of  the  liquid.  The  truth  of  this  principle  may  be  seen  if  we 
imagine  the  body  removed  from  the  liquid  and  the  space  which 
it  occupied  filled  with  the  liquid.  The  whole  liquid  mass  will 
then  be  in  equilibrium,  and  the  weight  of  the  portion  which 
has  been  introduced  will  be  equal  and  opposite  to  the  resultant 
of  the  pressures  applied  to  its  surface  by  the  surrounding 
liquid.  Exactly  these  same  pressures,  having  the  same  re- 
sultant, were  applied  to  the  body  when  it  w7as  in  the  liquid, 


04  MECHANICS    OF    LIQUIDS. 

and  its  apparent  weight  is  therefore  less  than  its  real  weight 
bv  an  amount  equal  to  this  resultant  force,  or  to  the  weight 
of  the  liquid  which  it  displaces. 

The  method  which  Archimedes  is  said  to  have  used  in  the 
t  actual  solution  of  the  problem  of  the  crown  will  illustrate  this 
principle.  The  apparatus  which  he  employed  was  an  equal 
armed  lever  or  balance,  on  one  arm  of  which  a  sliding  weight 
could  be  moved.  A  weight  of  gold  equal  to  the  weight  of  the 
crown  was  hung  on  one  end  of  this  balance  and  an  equal  weight 
of  silver  on  the  other.  The  masses  of  gold  and  silver  were  next 
immersed  in  water.  The  balance  was  then  no  longer  in  equi- 
librium, the  gold  appearing  to  be  heavier  than  the  silver. 
Equilibrium  was  restored  by  placing  the  sliding  weight  at  a 
certain  distance  from  the  point  of  suspension.  It  is  plain, 
from  Archimedes'  principle,  that  the  moment  of  force  intro- 
duced by  this  weight  was  equal  to  the  moment  of  force  in 
the  opposite  sense  introduced  by  the  excess  of  upward  pressure 
exerted  by  the  water  on  the  silver  over  that  exerted  on  the 
gold.  This  moment,  therefore,  was  proportional  to  the  excess 
of  volume  of  the  silver  over  that  of  the  gold.  The  crown  was 
next  substituted  for  the  silver,  and  the  gold  and  the  crown 
immersed  in  water.  Since  the  crown  contained  silver,  its 
volume  was  greater  than  that  of  the  gold,  and  there  was  an 
excess  of  upward  pressure  on  it.  Equilibrium  was  restored  by 
placing  the  sliding  weight  at  another  distance  from  the  axis 
of  suspension.  By  reasoning  similar  to  that  already  employed, 
we  conclude  that  the  moment  of  force  introduced  by  the  weight 
in  this  case  was  proportional  to  the  excess  of  the  volume  of 
the  crown  over  that  of  the  gold.  This  excess  was  due  to  the 
silver  in  the  crown,  and  proportional  to  the  amount  of  silver 
present,  so  that  the  ratio  of  the  quantity  of  silver  in  the  crown 
to  the  quantity  of  silver  which  weighed  as  much  as  the  crown, 
was  equal  to  the  ratio  of  the  moments  of  force  observed  in  the 
two  experiments,  or  to  the  ratio  of  the  arms  to  which  the 
sliding  weight  was  applied. 

46.  Floating  Bodies. — When  a  body  is  so  large  and  weighs 
so  little  that  it  displaces,  when  completely  submerged,  a  weight 


MECHANICS   OF   LIQUIDS.  65 

of  liquid  greater  than  its  own  weight,  it  will  not  sink  in  the 
liquid.  It  will  only  be  immersed  so  far  that  the  weight  of  the 
water  which  it  displaces  is  exactly  equal  to  its  own  weight. 
It  then  floats  upon  the  liquid.  When  such  a  body  is  placed 
upon  the  surface  of  the  liquid  it  will  generally  not  remain  at 
rest,  but  will  turn  round  until  it  attains  some  definite  position 
of  equilibrium.  If  it  is  afterwards  slightly  disturbed  from 
this  position,  it  will  come  back  to  it  again  if  its  equilibrium  is 
stable.  When  it  is  in  equilibrium,  its  weight  applied  at  the 
centre  of  gravity  and  the  resultant  of  the  upward  pressures 
which  are  applied  to  its  surface  must  be  equal  and  opposed  to 
each  other  in  the  same  vertical  line.  The  centre  of  gravity 
of  the  liquid  which  is  displaced  by  the  body  is  the  point  at 
which  the  resultant  of  the  upward  pressures  may  be  considered 
as  applied.  The  vertical  line  joining  the  centre  of  gravity  of 
the  body  and  the  centre  of  gravity  of  the  displaced  liquid  is 
called  the  axis  of  the  body.  If  the  axis  is  given  an  infinitesi- 
mal inclination,  the  point  at  which  it  is  intersected  by  the 
resultant  of  the  upward  pressures  is  called  the  metacentre. 
The  body  is  in  stable  equilibrium  when  the  metacentre  lies 
above  its  centre  of  gravity.  In  this  case,  when  the  body  is 
inclined,  its  weight  and  the  resultant  of  the  upward  pressures 
combine  to  form  a  couple  which  turns  the  body  back  into  its 
original  position. 

47.  Specific  Gravity  or  Relative  Density  .—If  we  weigh 
equal  volumes  of  diiferent  substances  we  find  that  in  general 
their  weights  are  different.  When  the  volume  which  is  chosen 
is  some  standard  volume,  these  different  weights  are  charac- 
teristic of,  or  specify,  the  different  substances.  By  the  aid  of 
Archimedes'  principle  it  is  easy  to  compare  the  weight  of  a 
body  with  the  weight  of  an  equal  volume  of  water,  taken  as 
a  standard  substance.  The  ratio  of  the  weight  of  the  body  to 
the  weight  of  an  equal  volume  of  water  is  a  characteristic 
number  for  the  substance  of  which  the  body  is  composed.  It 
is  therefore  called  the  specific  gravity  of  that  substance. 

The  mass  of  a  substance  which  is  contained  in  unit  volume 
is  called  the  density  of  that  substance.  From  the  proportion- 


66  MECHANICS   OF   LIQUIDS. 

ality  which  exists  between  mass  and  weight,  the  specific  grav- 
ity of  a  substance  is  the  ratio  of  its  density  to  the  density  of 
the  standard  substance  water.  Specific  gravity  is  therefore 
also  called  relative  density. 

When  the  metric  system  of  weights  and  measures  was 
introduced,  the  attempt  was  made  by  Borda  to  construct  the 
standard  mass,  or  kilogramme,  equal  to  the  mass  of  a  cubic 
decimetre  of  water  under  standard  conditions.  In  this  at- 
tempt he  succeeded  very  nearly,  so  that  unless  extraordinary 
accuracy  is  desired,  the  mass  of  a  quantity  of  water  in  kilo- 
grammes is  numerically  equal  to  the  volume  of  the  water  in 
cubic  decimetres.  The  gramme  is  the  mass  of  water  in  a  cubic 
centimetre.  In  the  system  of  units  which  we  employ,  in 
which  the  gramme  is  the  unit  of  mass,  the  centimetre,  the  unit 
of  length,  and  the  cubic  centimetre,  the  unit  of  volume,  the 
density,  defined  as  the  ratio  of  the  mass  to  the  volume,  is 
expressed  by  the  same  number  as  the  relative  density.  In 
other  systems  of  units  this  is  generally  not  the  case.  The 
relative  density,  which  is  a  simple  ratio  between  two  densi- 
ties, is  the  same  for  all  systems  of  units;  the  absolute  density 
is  different  in  the  different  systems. 

We  now  proceed  to  consider  some  of  the  methods  by  which 
the  relative  density  of  a  substance  may  be  determined. 

A  vessel  may  be  arranged  full  of  water,  so  that  the  water 
which  overflows  when  a  body  is  immersed  in  it  can  be  caught 
and  weighed.  The  volume  of  the  water  which  overflows  is 
equal  to  the  volume  of  the  body,  and  the  ratio  of  the  weight 
of  the  body  to  the  weight  of  the  water  is  therefore  the  rela- 
tive density. 

The  density  of  solids  which  are  insoluble  in  water  is  com- 
monly obtained  by  the  use  of  the  hydrostatic  balance.  This 
is  a  balance  so  constructed  that  the  body  to  be  tested  can 
hang  below  one  of  the  scale  pans.  The  body  is  first  weighed 
in  air.  It  is  then  immersed  in  water  and  its  apparent  weight 
determined.  By  Archimedes'  principle,  the  loss  in  weight 
which  it  apparently  undergoes  is  the  weight  of  an  equal  vol- 
ume of  water.  The  ratio  of  the  body's  weight  to  this  loss  in 


MECHANICS    OF    LIQUIDS.  67 

weight  is  therefore  the  relative  density  of  the  substance  of 
the  body. , 

The  relative  density  of  a  liquid  is  often  determined  by  an 
instrument  called  the  hydrometer.  Hydrometers  are  of  two 
kinds,  those  of  constant  weight,  and  those  of  constant  volume. 
The  hydrometer  of  constant  weight  is  generally  a  glass  bulb 
weigfiled  at  the  bottom,  and  furnished  above  with  a  long 
cylindrical  stem.  When  it  is  placed  in  a  liquid  it  stands  up- 
light  in  it,  Jind  a  proper  adjustment  of  its  wreight  will  cause 
it  to  sink  so  that  part  of  its  stem  is  immersed.  It  is  standard- 
ized by  placing  it  in  water,  and  marking  the  point  on  the  stem 
at  which  the  stem  protrudes  from  the  water  surface.  It  is 
then  placed  in  another  liquid  of  known  density,  and  the  place 
on  the  stem  again  marked  at  which  it  protrudes  from  the  sur- 
face of  the  liquid.  The  distance  between  these  two  points  is 
marked  off  into  any  arbitrary  number  of  equal  divisions,  and 
this  graduation  is  extended  above  and  below  the  standard 
points.  When  the  instrument,  thus  prepared,  is  placed  in  any 
liquid,  it  sinks  to  a  certain  point,  which  may  be  read  off  by 
means  of  the  graduation.  Its  weight  remaining  always  the 
same,  the  weight  of  the  liquid  which  it  displaces  is  always  the 
same,  and  since  the  point  to  which  it  sinks  determines  the 
volume  of  the  liquid  displaced,  we  have  the  means  of  determin- 
ing the  volumes  of  equal  weights  of  different  liquids,  and  so 
of  determining  their  relative  densities.  The  hydrometer  of 
constant  volume  is  a  glass  or  metal  bulb,  weighted  at  the 
bottom,  and  carrying  at  the  top  a  cylindrical  stem,  supporting 
a  pan  in  which  weights  may  be  placed.  The  instrument  is 
floated  in  a  liquid  and  weights  are  placed  in  the  pan  until  the 
instrument  sinks  to  a  marked  point  on  the  stem.  The  weight 
of  the  instrument  and  the  weights  in  the  pan  are  together 
equal  to  the  weight  of  the  liquid  displaced.  By  using  this 
instrument  in  different  liquids,  one  of  which  is  the  standard 
liquid,  we  may  determine  the  weights  of  equal  volumes  of  the 
liquids,  and  therefore  their,  relative  densities. 

The  rejative  density  of  liquids  is  often  determined  by  the 
use  of  the  specific  gravity  bottle,  or  pyknometer.  This  is  a 


68  MECHANICS    OF    LIQUIDS. 

small  bottle  or  flask  or  other  vessel  of  constant  volume.  The 
weight  of  water  which  will  fill  it  is  determined,  once  for  all. 
To  determine  the  relative  density  of  another  liquid,  it  is  filled 
with  that  liquid  and  the  weight  of  the  liquid  determined. 

48.  The  Barometer.—  The  attention  of  Galileo'  was  once 
^called  to  the  fact  that  the  water  of  a  certain  deep  well  could 
not  be  drawn  out  of  it  by  a  pump.  The  water  rose  in  the  pipe 
about  34  feet  and  could  not  be  raised  further  by  continued 
pumping.  At  that  time  the  rise  of  water  in  a  pump  was  con- 
sidered an  exhibition  of  a  general  principle,  embodied  in  the 
statement  "Nature  abhors  a  vacuum."  Galileo  recognized  that 
the  fact  which  had  been  called  to  his  attention  proved  at  least 
that  this  principle  was  limited  in  its  application,  but  he -was 
not  able  to  explain  it. 

Torricelli,  one  of  his  friends,  starting  with  the  knowledge 
which  he  had  obtained  from  Galileo  that  air  has  weight,  con- 
sidered it  to  be  a  simple  case  of  equilibrium  between  the 
pressure  of  the  water  column  in  the  pipe  and  of  the  air.  To 
verify  this  conclusion,  he  filled  a  long  glass  tube,  closed  at  one 
end,  with  mercury,  and  stopping  the  other  end  with  his  finger, 
inverted  the  tube  and  inserted  the  lower  end  of  it  in  a  vessel* 
of  mercury.  On  removing  his  finger,  the  column  of  mercury 
in  the  tube  fell  until  its  highest  point  was  about  30  inches 
above  the  surface  of  the  mercury  in  the  vessel  and  settled  to 
rest  in  that  position.  On  the  supposition  that  the  mercury 
column  is  sustained  by  the  pressure  of  the  air  on  the  free 
surface  of  the  mercury  in  the  vessel,  the  height  of  the  mercury 
column  ought  to  be  proportioned  to  the  height  of  the  water 
column,  in  the  experiment  with  the  pump,  inversely  as  the 
respective  masses  of  mercury  and  water  which  occupy  the  same 
volume.  For,  the  pressure  of  the  air  is  the  same  in  both  cases, 
and  therefore  the  pressures  of  the  two  columns  of  mercury 
and  water  should  be  the  same,  if  the  hypothesis  is  correct. 
The  general  expression  for  the  pressure  of  a  column  of  liquid 
may  be  found  by  considering  a  column  of  unit  cross  section 
and  of  height  h.  The  volume  of  such  a  column  is  also  ex- 
pressed by  h,  and  the  weight  of  it  is  equal  to  its  volume,  multi- 


MECHANICS   OF    LIQUIDS.  69 

plied  by  the  weight  contained  in  unit  volume.  This  product 
then  measures  the  pressure.  Two  liquid  columns  will  exert 
the  same  pressure  when  this  product  is  the  same  for  both  of 
them.  Their  heights  are  therefore  inversely  as  the  weights  of 
each  contained  in  unit  volume.  To  apply  this  to  the  case  under 
consideration  we  need,  in  addition  to  the  data  already  given, 
the  fact  that  the  weight  of  a  given  volume  of  mercury  is  13.6 
times  as  great  as  the  weight  of  an  equal  volume  of  water. 
Accordingly,  the  height  of  the  water  column  sustained  by  the 
pressure  of  the  air  should  be  13.6  times  as  great  as  the  height 
of  the  corresponding  mercury  column.  This  conclusion  from 
Torricelli's  hypothesis  was  verified  by  his  observations,  and 
he  accordingly  concluded  that  the  water  rose  in  the  pump  and 
that  the  mercury  was  sustained  in  the  tube  by  a  common 
cause,  the  pressure  of  the  air. 

The  arrangement  constructed  by  Torricelli  may  be  set  up 
permanently  as  a  means  of  measuring  the  pressure  of  the 
atmosphere.  It  is  then  called  a  barometer.  The  pressure  of 
the  atmosphere  is  found  to  vary  from  time  to  time,  but  it 
never  differs  very  much  at  any  one  place  from  a  mean  or 
standard  pressure  for  that  place. 

Pascal  contributed  to  the  verification  of  Torricelli's 
hypothesis  by  showing  that  the  height  of  the  mercury  column 
in  the  barometer  diminishes  when  the  barometer  is  carried  up 
a  mountain.  The  reason  for  this  is  plain  when  we  notice  that 
the  barometer  at  the  higher  station  is  relieved  from  the 
pressure  of  a  column  of  air,  whose  height  is  the  difference  of 
level  between  the  lower  and  upper  stations. 

The  pressure  indicated  by  the  barometer  under  standard 
conditions  is  often  used  as  a  unit  of  pressure.  This  unit  of 
pressure  is  called  the  pressure  of  one  atmosphere,  or  some- 
times the  atmo.  It  is  the  pressure  indicated  by  the  barometer 
when  placed  at  the  sea-level  and  when  the  height  of  the  column 
is  760  millimetres. 

49.  Flow  of  Liquids.  Torricelli's  Theorem. — It  was  ob- 
served by  Torricelli  that  when  a  stream  of  water  issues  from 


70  MECHANICS   OF    LIQUIDS. 

a  small  orifice  in  the  side  of  a  vessel  full  of  water,  the  velocity 
of  the  stream  is  the  same  as  that  which  a  body  would  acquire 
by  falling  from  the  surface  of  the  water  to  the  orifice.  This 
relation  is  called  Torricelli's  theorem.  It  may  be  examined  by 
fitting  to  the  orifice  a  short  tube  so  bent  that  the  stream 
issuing  from  it  is  directed  upwards,  in  which  case  it  will  be 
found  that  the  stream  rises  to  the  same  level  as  that  of  the 
water  in  the  vessel.  It  may  also  be  examined  by  allowing  the 
stream  to  issue  horizontally  and  determining  the  dimensions 
of  the  parabola  which  it  describes. 

When  we  attempt  another  verification  of  Torricelli's  state- 
ment, by  a  determination  of  the  quantity  of  water  which 
issues  in  a  given  time,  we  obtain  results  inconsistent  with  it. 
The  amount  of  water  obtained  is  always  less  than  that  which 
should  issue  from  an  orifice  of  the  given  size,  if  the  velocity 
is  that  stated  in  Torricelli's  theorem.  Newton  observed  that, 
owing  to  the  way  in  which  the  parts  of  the  water  near  the 
orifice  rush  together  as  they  issue  from  it,  the  diameter  of  the 
stream  outside  the  orifice  is  not  as  great  as  that  of  the  orifice. 
The  narrow  portion  of  the  stream  is  called  the  vena  contracta. 
When  the  quantity  of  water  which  issues  is  calculated  as  if  it 
were  issuing  from  an  orifice  whose  diameter  is  that  of  the 
vena  contracta,  the  results  obtained  are  consistent  with  the 
calculation. 

When  a  liquid  fiows  through  a  channel  or  series  of  pipes, 
whose  cross-sections  are  different  at  different  places,  the 
velocity  of  the  liquid  will  vary  along  the  channel,  being  at 
different  points  inversely  as  the  cross-sections  at  those  points. 

When  a  columnof  liquid  is  moving  along  a  pipe,  the  pressure 
of  the  liquid  outward,  against  the  wall  of  the  pipe,  is  dimin- 
ished by  an  amount  which  depends  upon  the  velocity  of  the 
liquid.  This  diminution  of  pressure  is  theoretically  propor- 
tional to  the  square  of  the  velocity.  The  velocity  may  there- 
fore be  so  great  that  the  pressure  against  the  wall  of  the  pipe 
disappears,  in  which  case  the  liquid  will  flow  past  an  orifice 
made  in  the  side  of  the  pipe,  without  issuing  from  it.  If  the 
velocity  is  still  further  increased,  a  negative  pressure,  or  one 


MECHANICS   OF    LIQUIDS.  71 

directed  inwards,  will  be  set  up,  the  air  will  enter  through  the 
orifice  from  without  and  will  be  carried  along  with  the  liquid. 

50.  Waves  on  the  Surface  of  a,  Liquid. — When  a  liquid 
surface  is  disturbed,  the  disturbance  passes  outward  in  every 
direction  as  a  wave  or  a  series  of  waves.  Each  of  these  waves 
consists  of  a  portion  of  the  liquid  which  is  elevated  above  the 
general  level,  and  another  portion  which  is  depressed  below  it. 
Our  common  observation  of  the  movements  of  floating  bodies 
when  waves  pass  under  them,  shows  that  the  liquid  is  not  car- 
ried along  with  the  waves.  The  wave  is  therefore  simply  a 
mode  of  motion  which  is  impressed  successively  upon  the  dif- 
ferent parts  of  the  surface.  We  may  study  the  motion  of  the 
liquid  caused  by  the  waves,  by  immersing  in  it  small  fragments 
of  some  solid  whose  relative  density  is  the  same  as  that  of  the 
liquid.  In  the  case  of  water  small  fragments  of  amber  may  be 
used.  If  a  succession  of  similar  waves  passes  over  the  surface, 
the  fragments  in  the  surface  describe  circles,  whose  diameters 
are  equal  to  the  vertical  distance  between  the  highest  point 
and  the  lowest  point  of  the  wave.  At  the  highest  point  of 
the  wave  they  move  forward  in  the  direction  in  which  the  wave 
is  travelling.  At  the  lowest  point  they  move  backward  with 
the  same  velocity.  The  fragments  which  are  below  the  surface 
describe  ellipses,  which  become  smaller  and  smaller,  and  more 
and  more  excentric  as  the  depth  increases,  until  at  last  they 
cannot  be  distinguished  from  minute  horizontal  lines. 

The  disturbance  thus  described  does  not  penetrate  far  be- 
low the  surface,  and  when  the  depth  of  the  liquid  is  very  great, 
most  of  it  remains  quiet,  even  when  the  waves  which  pass  over 
it  are  large.  When  the  depth  of  the  liquid  is  not  great,  the 
backward  part  of  the  motion,  which  is  in  the  lower  part  of 
the  curves  described  by  the  parts  of  the  liquid,  is  retarded, 
and  the  forward  motion  is  in  excess.  The  top  of  the  wave  thu| 
moves  forward  faster  than  the  bottom,  and  if  the  wave  ad- 
vances in  such  a  direction  that  the  depth  continually  decreases, 
it  will  at  last  curl  over  and  break. 


72  GRAVITATION   OB   MASS   ATTRACTION. 


GRAVITATION   OR   MASS   ATTRACTION. 

51.  Kepler's  Laws  of  Planetary  Motion. — By  an  extended 
study  of  the  apparent  motions  of  the  planet  Mars,  and  then 
of  those  of  the  other  planets,  Kepler  showed  (1609-1619)  that 
they  could  be  represented  by  supposing  the  motions  of  the 
planet  to  conform  to  the  following  laws,  which  are  known  as 
Kepler's  laws: 

1.  The  path  of  a  planet  is  an  ellipse,  at  one  of  the  foci  of 
which  the  sun  is  situated. 

2.  The  radius  vector  drawn  from  the   sun   to   a   planet 
sweeps  out  equal  areas  in  equal  times. 

3.  The  squares  of  the  periodic  times  of  the  planets  are 
proportional  to  the  cubes   of  the  semi-major   axes   of  their 
elliptic  orbits. 

By  the  periodic  time  of  a  planet  is  meant  the  time  required 
by  it  to  completely  describe  its  orbit. 

52.  The  Law  of  Gravitation  or  Mass  Attraction. — The  laws 
of  Kepler  merely  describe  motions  which  are  consistent  with 
the  apparent  motions  of  the  planets.    On  the  assumption  that 
they  describe  the  real  motions  of  the  planets,  it  became  a 
question  of  the  utmost  interest  to  determine  the  forces  which 
caused  the  motions.    This  question  was  first  answered  by  New- 
ton in  his  Principia  (1687).    Newton's  investigation  is  an  ex- 
cellent illustration,  perhaps  the  best  illustration   which   we 
have,  of  the  process  of  thought  employed  in  physical  reasoning, 
as  described  in  §  2. 

Newton  first  made  the  hypothesis  that  each  planet  is  acted 
on  by  a  force  which  is  proportional  to  its  mass,  is  directed 
toward  the  sun,  and  is  inversely  proportional  to  the  square  of 
the  distance  between  the  planet  and  the  sun.  He  then  applied 
the  principles  of  mechanics  to  determine  what  would  be  the 
motion  of  a  body  acted  on  by  a  force  directed  toward  a  fixed 
point  according  to  this  hypothesis. 


GRAVITATION    OR    MASS    ATTRACTION.  73 

He  showed  that  if  the  force  which  acts  on  the  body  is 
directed  toward  a  fixed  point,  whatever  may  be  the  way  in 
which  the  force  varies  with  the  distance  of  the  body  from 
that  point,  the  radius  vector  drawn  from  the  point  to  the 
body  will  sweep  out  equal  areas  in  equal  times.  That  is,  a 
body  acted  on  by  a  force  which  is  always  directed  toward  a 
fixed  centre  conforms  to  Kepler's  second  law.  Hence  the  con- 
clusion may  be  drawn  that  the  force  which  acts  on  the  planet  is 
directed  toward  the  sun. 

Newton  showed  further  that  when  the  force  which  acts  on 
the  body  varies  inversely  with  the  square  of  the  distance  be- 
tween the  body  and  the  fixed  centre,  and  is  directed  toward 
that  centre,  the  path  of  the  body  will  be  one  of  the  conic  sec- 
tions. Whether  the  path  of  the  body  is  an  ellipse,  a  parabola, 
or  an  hyperbola  depends  on  the  velocity  of  the  body  as  it 
passes  the  end  of  the  major  axis.  When  this  velocity  does 
not  exceed  a  certain  limit,  the  path  of  the  body  will  be  an 
ellipse.  The  motion  of  the  body,  therefore,  which  is  caused  by 
the  hypothetical  force  supposed  to  act  on  it,  conforms  to  Kep- 
ler's first  law.  Hence  the  conclusion  may  be  drawn  that  the 
force  which  acts  on  a  planet  toward  the  sun  varies  inversely 
with  the  square  of  the  distance  between  the  planet  and  the 
sun.  , 

Newton  showed  further  that  when  forces  directed  toward 
a  common  centre  act  on  different  bodies  so  as  to  make  them 
describe  elliptic  orbits,  the  squares  of  the  times  in  which  they 
describe  their  orbits  are  proportional  to  the  cubes  of  the  semi- 
major  axes  of  the  orbits,  if  only  the  force  which  acts  on  each 
body  is  proportional  to  the  mass  of  the  body.  That  is,  if  the 
forces  which  act  upon  the  bodies  are  proportional  to  their 
masses,  the  motions  of  the  bodie«  will  conform  to  Kepler's 
third  law.  Hence  the  conclusion  may  be  drawn  that  the  forces 
which  act  vipon  the  planets  are  proportional  to  their  masses. 

The  results  of  the  preceding  paragraphs,  which  may  be 
summed  up  in  the  statement  that  the  planets  move  as  if  they 
were  attracted  toward  the  sun  by  forces  which  are  propor- 
tional to  their  masses  and  which  vary  inversely  with  the 


74  GKAVITATION    OK    MASS    ATTRACTION. 

squares  of  their  distances  from  the  sun,  are  all  that  can  be 
reached  by  the  study  of  the  motion  of  the  planets  as  expressed 
in  Kepler's  laws.  More  might  have  been  learned  from  a  study 
of  the  perturbations  caused  in  the  orbit  of  one  planet  by  the 
action  of  the  others  around  it.  Newton,  however,  did  not 
proceed  alone  this  line  in  his  argument,  but  followed  another 
course.  He  showed  that  the  motion  of  the  moon  conforms  to 
Kepler's  first  and  second  laws  when  the  earth  is  taken  as  the 
centre  of  force,  and  therefore  concluded  that  the  moon  is  acted 
on  by  a  force  directed  toward  the  earth  and  varying  inversely 
with  the  square  of  the  distance  of  the  moon  from  the  earth. 
He  then  showed,  from  a  calculation  of  the  dimensions  of  the 
moon's  orbit,  that  the  acceleration  of  the  moon  toward  the 
earth  is  equal  to  the  acceleration  of  a  body  near  the  earth's 
surface  diminished  in  the  inverse  ratio  of  the  squares  of  the 
distances  of  the  moon  and  of  the  body  from  the  centre  of  the 
earth.  From  this  the  conclusion  was  drawn  that  the  force 
whicli  acts  on  the  moon  and  determines  its  motion  in  its  orbit 
is  of  the  same  nature  as  the  force  which  causes  heavy  bodies 
to  fall  toward  the  earth;  and  further,  that  the  forces  which 
act  between  the  different  bodies  of  the  planetary  system  are 
of  the  same  nature.  This  force  is  therefore  called  the  force 
of  gravitation. 

Newton  constructed  two  pendulums  of  equal  length  whose 
bobs  were  two  similar  spherical  boxes,  in  which  different 
bodies  could  be  placed.  He  found  that  whatever  were  the  sub- 
stances placed  in  the  boxes,  the  times  of  oscillation  of  the 
pendulums  were  the  same,  and  hence  concluded  that  the  force 
of  gravity  is  proportional  exactly  to  the  mass  of  the  body 
acted  on.  and  is  independent  of  its  other  characteristics. 

Observations  of  the  motions  of  Jupiter's  satellites,  as  well 
as  of  the  satellites  of  the  other  planets,  show  that  they  con- 
form to  Kepler's  third  law  when  the  planet  is  taken  as  the 
centre  of  force,  and  hence  that  the  forces  which  act  on  them 
toward  the  planet  are  proportional  to  their  masses.  On  the 
assumption  that  the  force  with  which  the  planet  acts  on  the 
sun  follows  the  same  law,  the  attraction  between  the  sun  and 


GRAVITATION   Oli    MASS   ATTRACTION.  76 

the  planet  is  proportional  to  the  mass  of  the  planet  and  also 
to  the  mass  of  the  sun. 

The  arguments  which  have  now  been  adduced  lead  to  the 
general  conclusion  that  a  force  acts  between  any  two  portions 
of  matter  in  the  universe,  which  tends  to  draw  them  together, 
and  that  the  magnitude  of  this  force  is  proportional  to  each 
of  the  masses  and  inversely  proportional  to  the  square  of  the 
distance  between  them.  This  law  is  called  the  law  of  gravita- 
tion or,  from  the  fact  that  the  force  depends  directly  on  the 
masses  of  the  interacting  bodies,  the  law  of  mass  attraction. 
If  we  represent  by  m  the  mass  of  one  of  the  bodies,  by  m',  the 
mass  of  the  other,  by  r,  the  distance  between  them  and  by  k, 
a  constant  or  factor  of  proportion,  we  may  express  the  force 

of  gravitation  F  between  these  bodies  by  the  formula  F  =  kmm  • 

The  factor  of  proportion  fc  is  the  same  in  all  cases. 
«^"  53.  Gravitation  Constant. — The  factor  of  proportion  which 
Tias  been  introduced  in  the  formula  just  given  is  called  the 
gravitation  constant.  It  is  manifestly  equal  to  the  force  which 
two  unit  masses  will  exert  on  each  other  when  they  are  unit 
distance  apart.  It  may  be  determined  by  a  direct  determina- 
tion of  the  gravitational  force  with  which  two  bodies,  whose 
masses  are  known,  act  on  each  other. 

An  experiment  designed  for  this  purpose  by  Mich  ell  was 
carried  out  by  Cavendish  in  1798.  It  is  always  known  and 
referred  to  as  Cavendish's  experiment.  A  horizontal  rod,  carry- 
ing on  its  ends  two  equal  balls  of  lead,  was  suspended  at  its 
middle  point  from  a  long  cylindrical  wire.  When  not  acted 
on  by  any  other  forces  than  its  own  weight,  such  a  system  will 
assume  a  definite  position,  in  which  there  is  no  twist  in  the 
wire.  If  the  wire  is  twisted  by  turning  the  rod  through  any 
angle  in  the  horizontal  plane,  the  twist  in  the  wire  will  intro- 
duce an  elastic  reaction  proportional  to  the  angular  displace- 
ment of  the  rod,  and  by  preliminary  experiments  the  magnitude 
of  the  couple  which  will  cause  a  given  twist  in  the  wire  may 
be  determined.  After  such  a  determination  had  been  made, 
two  large  balls  of  lead  were  so  placed  on  either  side  of  the  rod 


76  GRAVITATION    OR    MASS    ATTRACTION. 

as  to  attract  the  small  balls  in  opposite  directions,  and  so  to 
introduce  a  couple  which  twisted  the  wire.  The  magnitude  of 
this  couple  was  measured  for  different  distances  between  the 
large  and  small  balls,  and  thus  the  forces  which  they  exerted 
on  each  other  were  determined.  Cavendish  found  that  the 
force  between  the  balls  varied  inversely  as  the  square  of  the 
distance  between  their  centres,  and,  by  using  balls  of  different 
masses,  that  the  force  between  them  was  proportional  to  the 
masses.  The  values  of  the  forces  thus  determined,  introduced, 
with  the  known  masses  of  the  balls  and  the  distance  between 
their  centres,  into  the  formula  expressing  the  law  of  gravita- 
tion, lead  to  the  determination  of  the  gravitation  constant. 
The  Cavendish  experiment  has  been  several  times  repeated, 
with  every  precaution  to  insure  accuracy.  The  results  obtained 
by  different  observers  are  fairly  consistent  with  each  other. 
The  value  of  the  gravitation  constant  obtained  from  them, 
when  the  unit  of  mass  is  the  gramme  and  the  unit  of  distance 
the  centimetre,  is  about  sixty-five  billionths  of  a  dyne. 

A  method  recently  employed  by  von  Jolly  led  to  the  same 
result.  In  this  method  a  sensitive  balance  was  mounted  high 
in  a  tower  and  additional  scale  pans  were  suspended  from  it 
by  long  wires.  This  balance  was  first  used  to  show  that  the 
weight  of  a  body  varies  with  its  distance  from  the  earth's 
surface.  A  weight  placed  in  one  of  the  upper  scale  pans  was 
balanced  by  another  weight  in  the  other  upper  scale  pan.  This 
second  weight  was  then  placed  in  the  lower  scale  pan  and 
found  to  be  heavier  than  before.  Its  increase  in  weight  was 
determined,  and  measured  the  increase  in  the  attraction  of 
the  earth  upon  it,  due  to  its  being  brought  nearer  the  earth's 
centre.  To  determine  the  gravitation  constant,  a  large  block 
of  lead  was  placed  under  the  weight  in  the  lower  scale  pan, 
which,  by  its  attraction,  caused  a  further  increase  in  that 
weight.  This  was  carefully  measured,  and  from  it,  taken 
together  with  the  known  values  of  the  attracting  masses  and 
their  dimensions,  the  gravitation  constant  was  calculated. 

54.  Acceleration  at  the  Earth's  Surface  Due  to  Gravity. — 
In  studying  the  motions  of  bodies  caused  by  the  earth's  at- 


GRAVITATION    OR    MASS    ATTRACTION.  77 

traction  we  may  consider  the  earth  as  fixed,  the  movement  of 
the  earth  toward  the  common  centre  of  mass,  which  occurs 
when  the  body  moves  toward  the  earth,  being  so  small  that  it 
may  be  neglected.  This  being  so,  it  follows  from  the  law  of 
gravitation  that  all  bodies  near  the  earth's  surface  will  have 
the  same  acceleration.  As  has  already  been  seen,  this  con- 
clusion was  reached  by  Galileo  from  observation.  This  ac- 
celeration is  represented  by  g.  It  is  of  great  practical  im- 
portance to  know  its  exact  value,  because  it  furnishes  us  the 
most  precise  measure  of  force  in  absolute  units.  We  can 
determine  the  mass  of  a  body  very  exactly,  and  use  its  weight 
as  a  force.  Then  the  force  exerted  by  the  body,  expressed  in 
absolute  units,  is  its  mass  multiplied  by  this  acceleration. 

Bodies  fall  so  fast  that  it  is  extremely  difficult  to  make 
exact  observations  on  them.  A  method  has  been  devised  for 
this  purpose,  depending  on  the  use  of  electric  currents  and 
magnets,  which  yields  fairly  accurate  values  of  g.  Observa- 
tions on  bodies  rolling  down  an  inclined  plane,  or  with  the 
Attwood's  machine,  will  yield  approximate  values  of  this 
constant.  Its  exact  value  is  obtained  universally  by  means  of 
the  pendulum,  first  used  for  this  purpose  by  Huygens.  The 
pendulum  is  used  for  this  purpose  in  two  forms.  In  one  of 
these,  known  as  Borda's  pendulum,  the  bob  is  a  heavy  sphere 
and  the  suspension  a  long  cylindrical  wire.  The  wire  is  hung 
from  a  knife-edge,  which  rests  on  agate  planes.  The  effect  of 
the  knife-edge  on  the  period  of  the  pendulum  is  avoided  by  a 
preliminary  adjustment.  The  efficient  parts  of  the  pendulum 
are  therefore  regular  bodies  and  the  moment  of  inertia  and 
the  static  moment  of  the  pendulum  can  be  calculated.  The 
period  of  oscillation  is  observed,  and,  by  the  aid  of  the  formula 
given  in  §40,  the  value  of  g  is  determined. 

In  the  other  form,  known  as  Kater's  pendulum,  the  pendu- 
lum is  a  stiff  rod  or  bar  carrying  a  heavy  weight  at  one  end 
and  furnished  with  two  knife-edges  which  confront  each  other. 
An  additional  movable  weight  is  also  carried  on  the  bar.  By 
a  suitable  adjustment  of  this  weight,  the  periods  of  the  pendu- 
lum swinging  about  the  two  knife-edges  may  be  made  the 


78  GRAVITATION    OR    MA£S   ATTRACTION. 

same.  The  pendulum  thus  adjusted  is  a  reversible  pendulum, 
and,  as  was  seen  in  the  discussion  of  §40,  the  distance  between 
the  knife-edges  is  the  length  of  the  simple  pendulum  which 
will  swing  in  the  same  period.  Having  measured  this  distance 
once  for  all,  and  having  determined  the  period,  the  value  of  g 
may  be  calculated  by  the  aid  of  the  formula  for  the  simple 
pendulum. 

The  value  of  g  is  different  at  different  places,  ranging  be- 
tween 978  at  the  equator  and  983  at  the  pole,  when  the  units 
of  length  and  time  are  the  centimetre  and  second.  For  ordi- 
nary calculations  in  our  latitude  we  may  use  the  value  980. 

55.  Density  of  the  Earth. — Newton  did  not  attempt  any 
experimental  demonstration  of  his  hypothesis  that  an  at- 
traction exists  between  any  two  bodies.  The  experiment  of 
Cavendish  by  which  this  hypothesis  was  confirmed  was  made 
long  after  his  time.  The  attempts  which  were  made  to  confirm 
it,  before  Cavendish's  experiment  was  executed,  involved  a 
comparison  of  the  attraction  of  the  earth  for  a  body  with  the 
attraction  exerted  on  that  body  by  a  portion  of  the  earth,  and 
the  numerical  results  obtained  were  worked  over  so  that  the 
result  stated  was  a  value  of  the  mean  density  of  the  earth. 

The  first  of  these  experiments  was  made  by  the  astronomer 
Maskelyne  in  1774.  The  theory  of  this  experiment  is  as  follows: 
If  two  stations  on  the  same  meridian  are  chosen  and  the  lati- 
tudes of  these  stations  observed,  the  difference  of  latitude  will 
plainly  be  equal  to  the  angle  between  the  radii  of  the  earth 
drawn  from  its  centre  to  the  two  stations,  if  the  plumb-lines 
with  reference  to  which  the  latitudes  are  measured  are  in  the 
directions  of  those  radii.  If,  however,  a  great  mass,  like  a 
mountain,  stands  between  the  two  stations,  the  plumb-bob 
south  of  it  will  be  drawn  toward  the  north  so  that  the  latitude 
measured  at  that  station  will  be  less  than  the  true  latitude. 
The  plumb-bob  north  of  the  mountain  will  be  drawn  toward 
the  south  so  that  the  latitude  measured  at  the  northern  station 
will  be  greater  than  the  true  latitude.  If  we  know  the  true 
difference  of  latitude  between  the  two  stations,  from  a  direct 
measurement  of  their  distances  apart  and  our  knowledge  of 


GKAV1TATION    OR    MASS    ATTRACTION.  79 

the  size  of  the  earth,  we  may  determine,  by  the  observations 
here  indicated,  the  amount  by  which  the  plumb-bob  has  been 
drawn  aside  by  the  attraction  of  the  mountain,  and  hence  may 
determine  the  relative  values  of  the  attraction  of  the  mountain 
and  of  that  of  the  earth.  We  may  then  determine  the  actual 
mass  of  the  mountain  from  its  dimensions  and  the  average 
density  of  its  parts,  obtained  from  a  study  of  specimens  of 
the  materials  which  compose  it.  From  these  data,  assuming 
the  law  of  gravitation,  we  may  calculate  the  mass  of  the  earth 
and  hence  its  average  density.  By  an  experiment  like  the  one 
here  outlined,  Maskelyne  obtained  for  the  mean  density  of  the 
earth  the  value  4.7. 

Another  essentially  similar  method  was  employed  by  Car- 
lini  in  1824.  He  observed  the  oscillations  of  a  pendulum -at 
the  bottom  and  at  the  top  of  a  mountain.  Assuming  the  law 
of  gravitation,  he  calculated  what  the  acceleration  should  be 
at  the  top  of  the  mountain  if  its  difference  from  that  at  the 
bottom  depended  solely  on  the  change  of  level,  and  compared 
this  calculated  result  with  the  acceleration  observed  at  the 
top  of  the  mountain.  He  found  that  the  observed  acceleration 
was  greater  than  the  calculated  one.  The  difference  he 
ascribed  to  the  attraction  of  the  mountain.  By  proceeding 
from  this  point  as  in  Maskelyne's  experiment,  he  obtained  for 
the  density  of  the  earth  the  value  4.8. 

The  experiments  of  Cavendish  and  of  von  Jolly  may  also 
be  used  in  the  determination  of  the  earth's  density.  The  re- 
sults obtained  in  these  experiments  are  much  more  consistent 
with  each  other  that  those  obtained  by  methods  in  which  the 
attraction  of  large  parts  of  the  earth  is  observed,  and  may  be 
received  with  greater  confidence.  The  value  of  the  mean 
density  of  the  earth  obtained  from  them  is  about  5.64. 


80  KLAST1CITY. 


ELASTICITY. 

5G.  Cohesion. — When  we  try  to  break  apart  a  piece  of  metal 
or  wood,  we  encounter  a  very  considerable  resistance.  This 
resistance  is  a  manifestation  of  forces  which  act  between  the 
parts  of  the  body.  If  we  make  the  supposition  that  the  mat- 
ter of  the  body  is  so  distributed  as  to  occupy  a  considerable 
part  of  its  volume,  it  may  be  shown  by  analysis  that  these 
forces  cannot  be  due  to  the  attraction  of  gravitation  between 
the  parts,  the  forces  arising  from  gravitation  being  so  small 
as  to  be  negligible  in  comparison  with  them.  The  parts  of  the 
bddy  cohere  with  each  other,  and  the  forces  between  them  are 
called  forces  of  cohesion.  They  may  be  considered  attractions 
between  the  parts  of  the  body. 

When  we  attempt  to  compress  the  body  into  a  smaller  vol- 
ume, we  again  encounter  a  resistance,  which  may  be  ascribed 
to  a  repulsion  between  the  parts  of  the  body.  It  is  not  cer- 
tain that  these  repulsive  forces  really  exist.  The  effects 
ascribed  to  them  may  also  be  ascribed  to  the  motions  of  the 
parts  of  the  body,  due  to  the  heat  of  the  body.  We  shall, 
however,  adopt  this  hypothesis  of  repulsive  forces  provision- 
ally. If  we  do  so,  we  must  consider  the  form  and  size  of  a 
body,  in  any  condition  in  which  its  parts  are  relatively  at  rest, 
to  be  determined  by  a  balance  between  the  attractions  and 
repulsions  in  every  part  of  it. 

For  a  reason  which  will  appear  hereafter  these  forces  be- 
tween the  parts  of  the  body  are  often  called  molecular  forces. 

57.  Elasticity. — If  a  metallic  wrire  is  fastened  at  one  end, 
and  a  weight  hung  on  the  other,  the  wire  will  stretch  a  little. 
The  addition  of  another  weight  will  stretch  it  further.  When 
the  weights  are  removed,  provided  the  wire  has  not  stretched 
beyond  a  certain  limit,  its  length  will  become  what  it  was  at 
first.  Phenomena  essentially  similar  are  exhibited  by  a  rod 
when  it  is  bent,  a  wire  when  it  is  twisted,  a  column  on  top  of 
which  a  heavy  weight  is  placed,  or  a  mass  of  water  or  air 


ELASTICITY.  81 

confined  in  a  vessel  and  subjected  to  pressure.  In  each  case 
of  this  sort,  the  forces  which  counteract  those  applied  to  the 
body  are  evidently  the  forces  of  cohesion  and  of  repulsion 
which  have  just  been  considered.  The  system  of  forces  or 
pressures  applied  to  the  body  may  be  called  in  general  a  stress. 
The  change  in  shape  or  size  or  in  both  which  the  body  under- 
goes may  be  called  its  deformation  or  strain.  As  a  result  of 
common  observation  we  may  say  that  a  stress  applied  to  a 
body  always  produces  a  strain  in  it.  If  the  body  recovers  its 
original  condition,  either  in  whole  or  in  part,  when  the  stress 
is  removed,  it  is  called  an  elastic  body.  The  forces  which  it 
exerts  against  the  applied  stress  are  called  elastic  forces,  and 
the  elasticity  of  the  body  is  the  property  which  it  possesses  of 
exerting  such  forces.  So  far  as  our  experiments  go,  there  is 
no  body  which  does  not  possess  elasticity,  if  not  with  respect 
to  all  stresses,  at  least  with  respect  to  certain  types  of  stress. 

58.  Hooke's  Laiv. — In  1679  Hooke  gave  an  account  of  ex- 
periments which  he  had  tried  on  the  stretching  of  wires  by 
weights.     He  found  that   for   wires   of  the   same   length  and 
thickness,  and  of  the  same  material,  the  elongations  produced 
by  the  addition  of  different  weights  were  proportional  to  those 
weights.    He  stated  this  result  of  his  observations  in  the  law, 
Ut  tensio,  sic  vis;  the  extension  is  proportional  to  the  force. 
This  law  is  of  far  wider  application  than  may  be  thought  from 
the  way  in  which  it  was  derived.     It  applies  to  all  cases  of 
strains  produced  by  stresses.     If  the  stress  is  of  a  certain 
type,  it  will  produce  a  strain  of  the  same  type.    Thus,  for  ex- 
ample, a  pull  on  a  wire  will  lengthen  it;  a  couple  applied  to 
one  end  of  a  wire,  whose  other  end  is  fixed,  will  twist  it;  a 
pressure  applied  to  a  mass  of  water  will  compress  it.    In  these 
cases,  and  in  all  others  which  might  be  cited,  the  law  holds 
that  the  strain  is  proportional  to  the  stress.    Hooke's  law  is 
therefore  the  general  law  of  elasticity. 

59.  Compression  and  Expansion. — A  pressure  applied  to  a 
surface  so  as  to  be  equal  at  every  point  on  that  surface,  and 
normal  to  it,  is  called  a  hydrostatic  pressure.    It  may  be  either 
positive   or  negative.     The   pressure   directed   outward   from 


82  ELASTICITY. 

the  surface  is  taken  as  positive;  that  directed  inward  is  nega- 
tive. The  bodies  which  we  consider,  in  our  present  discussion 
of  elasticity,  are  supposed  to  be  homogeneous  and  isotropic; 
that  is,  they  are  alike  in  every  part  and  possess  similar  prop- 
erties, with  respect  to  elastic  forces,  in  every  direction.  A 
hydrostatic  pressure  applied  to  such  a  body  will  change  its 
volume  without  changing  its  shape.  When  .the  pressure  is 
positive,  the  volume  of  the  body  increases,  or  the  body  under- 
goes expansion.  When  the  pressure  is  negative,  the  body 
undergoes  compression.  In  the  case  of  compressions  and  ex- 
pansions, Hooke's  law  may  be  stated  by  saying  that  the 
change  of  volume  of  the  body  is  proportional  to  the  change 
in  the  pressure  applied  to  it. 

The  change  in  volume  of  a  body,  caused  by  a  pressure,  is 
manifestly  proportional  to  the  number  of  units  of  volume  in 
the  body.  Bodies  of  different  sorts  exhibit  characteristic 
changes  of  volume  for  the  same  pressure  changes.  "When  the 
pressure  changes  by  one  unit,  the  decrease  or  increase  of  a 
unit  volume  will  measure  the  compressibility  or  expansibility 
of  the  substance  composing  the  body.  This  quantity,  while  it 
is  the  one  most  directly  open  to  experimental  examination,  is 
not  the  quantity  commonly  employed  to  represent  the  char- 
acteristics of  the  body  under  pressure.  We  use  instead  the 
ratio  of  the  increase  in  pressure  to  the  change  of  volume  of  a 
unit  volume.  This  quantity  is  called  the  volume  elasticity,  or 
often  simply,  the  elasticity,  of  the  substance  composing  the 
body. 

60.  Elasticity  of  Traction. — When  one  end  of  a  body  is 
fixed,  and  a  force  is  applied  to  the  other  end  to  lengthen  it  or 
to  shorten  it,  the  force  applied  is  called  a  traction.  The 
tractions  commonly  observed  are  pulls  which  lengthen  the 
body,  and  the  bodies  used  are  long  thin  ones,  like  wires  or  rods. 
For  such  a  body  Hooke's  law  takes  the  form  that  the  elonga- 
tion of  the  body,  that  is,  its  increase  in  length,  is  proportional 
to  the  stretching  force.  When  we  examine  wires  of  the  same 
material,  but  of  different  lengths  and  cross-sections,  we  find 
that  their  elongations  are  proportional  to  their  lengths,  and 


ELASTICITY.  83 

inversely  proportional  to  their  cross-sections.  Wires  of  equal 
lengths  and  cross-sections,  but  of  different  materials,  will 
exhibit  different  elongations  when  acted  on  by  equal  forces. 
The  preceding  statements  are  all  contained  in  the  formula 

e  =  _  ,  in  which  F  represents  the  force,   I,  the  length,   5,   the 

cross-section,   and  -,,     ihe   factor  of    proportion.      This    factor   is 

different  for  each  substance,  and  is  characteristic  of  it.  It  is 
evidently  equal  to  the  elongation  of  a  wire  of  unit  length  and 
unit  cross  section  stretched  by  a  unit  force.  Its  reciprocal  n 
is  more  commonly  employed  to  characterize  the  elasticity  of 
a  substance  with  respect  to  traction.  This  quantity  is  the 
ratio  of  the  force  applied  to  the  elongation  which  it  will 
produce  in  a  wire  of  unit  length  and  of  unit  cross-section.  It 
is  called  the  modulus  of  fractional  elasticity,  or  often  Young's 
modulus.  It  was  introduced  by  Thomas  Young  in  1807  as  a 
convenient  characteristic  number  for  elastic  bodies. 

61.  Elasticity  of  Torsion. — When  a  wire  clamped  at  one 
end  is  twisted  by  a  couple  applied  to  the  other  end,  a  pointer 
attached  to  it  in  the  plane  of  the  couple  will  turn  through  an 
angle  which  is  proportional  to  the  moment  of  couple.  The 
wire  is  said  to  undergo  torsion,  and  Hooke's  law  for  the  case 
of  torsion  is  given  by  the  statement  that  the  amount  of 
torsion  is  proportional  to  the  moment  of  couple.  Experiments 
on  wires  of  different  lengths  and  cross-sections  show  that  the 
amount  of  torsion  is  proportional  to  the  length  of  the  wire  and 
inversely  to  the  square  of  the  cross-section.  It  also  depends 
on  the  material  constituting  the  wire.  The  results  of  these 

experiments  are  collected  in  the  formula  tf>  =     *  ^   ,     in    which 

C  represents  the  moment  of  couple,  and  _?T  the  factor  of  pro- 
portion. The  quantity  n,  called  the  modulus  of  rigidity,  or 
the  rigidity,  is  different  for  different  substances,  and  character- 
istic of  them. 


84  ELASTICITY. 

02.  Elasticity  of  Flexure. — If  a  straight  rod  or  bar  is 
clamped  at  one  end,  and  if  the  other  end  is  pulled  aside  by 
any  force,  the  bar  is  bent  or  flexed.  The  amount  of  the  flex- 
ure, determined  by  the  distance  through  which  the  end  of  the 
bar  moves,  is  proportional  to  the  force  applied  to  produce  it. 
This  is  the  statement  of  Hooke's  law  for  this  case.  The 
amount  of  flexure  depends  also  on  the  dimensions  of  the  bar 
and  on  the  material  of  which  it  is  composed.  The  elastic  co- 
efficient or  modulus  characteristic  of  any  particular  substance 
in  this  case  is  the  same  as  the  modulus  of  tractional  elasticity. 
That  this  should  be  the  case  may  be  seen  if  we  consider  a  dia- 
gram representing  a  bent  rod.  The  line  running  down  the 
middle  of  the  rod  is"  not  changed  in  length  by  bending,  while 
those  parallel  to  it  above  it  are  elongated  and  tho~se  below  it 
shortened.  As  each  of  these  lines  reacts  against  the  bending 
force  in  a  way  that  depends  upon  its  elasticity  of  traction,  it 
is  plain  that  their  combined  effect  will  be  measured  in  terms 
of  that  elasticity. 

63.  Types  of  Strains  and  Stresses. — In  most  of  the  theo- 
retical discussions  of  the  properties  of  elastic  bodies  we  limit 
ourselves  to  the  consideration  of  very  small  deformations. 
We  suppose  these  deformations  to  be  such  that  a  line  in  the 
body  which  is  straight  before  the  deformation  remains 
straight  after  the  deformation. 

Analysis  shows  us  that  with  these  limitations  any  strain 
may  be  obtained  by  the  superposition  of  strains  of  two  types. 
These  two  types  of  strain,  called  compression  and  shear,  are 
distinctly  different. 

A  compression  is  a  change  of  volume,  of  such  a  sort  that 
the  proportions  of  the  parts  of  the  body  remain  the  same. 
That  is,  it  is  a  change  of  volume  without  a  change  of  shape. 
It  may  of  course  be  either  positive  or  negative.  A  shear  is  a 
change  of  shape  without  a  change  of  volume.  This  may  be 
brought  about  by  a  sliding  of  contiguous  planes  in  the  body 
over  each  other,  or  by  an  elongation  of  parallel  lines  in  the 
body  and  a  contraction  of  lines  in  some  one  direction  perpendicu- 
lar to  them. 


ELASTICITY.  85 

Corresponding  to  these  two  types  of  strain,  there  are  two 
types  of  stress,  the  hydrostatic  pressure  and  the  shearing 
stress.  The  hydrostatic  pressure  has  already  been  denned. 
The  shearing  stress  is  a  pair  of  equal  and  opposite  pressures 
applied  tangentially  to  parallel  surfaces  in  the  body,  combined 
with  another  pair  of  pressures  similarly  applied  to  two  other 
surfaces  in  the  body,  at  right  angles  to  the  first,  in  such  a 
way  that  the  system  of  four  pressures  introduces  no  moment 
of  couple  and  serves  merely  to  deform  the  body.  Correspond- 
ing to  the  two  fundamental  types  of  stress  and  the  two  funda- 
mental types  of  strain  which  they  produce,  we  have  the  two 
fundamental  moduli  of  elasticity,  the  elasticity  of  volume  and 
the  rigidity.  Other  moduli  of  elasticity,  in  particular  Young's 
modulus,  are  functions  of  these  fundamental  moduli. 

04.  Failure  of  Hooke's  Law. — No  mention  has  been  made, 
up  to  this  point,  of  the  fact  that  Hooke's  law  is  not  uni- 
versally true.  It  is  plainly  not  true  for  any  strain  whatever, 
because,  as  we  well  know,  a  body  to  which  sufficient  stress  is 
applied  may  be  permanently  deformed  or  broken.  When  ex- 
periments are  tried  with  stresses  of  different  magnitudes,  it 
is  found  that  Hooke's  law  holds  without  perceptible  error  for 
small  stresses,  but  fails  when  the  stress  exceeds  a  certain 
limit.  The  laws  which  have  been  stated  in  the  foregoing  sec- 
tion must  be  interpreted  in  accordance  with  this  statement. 

The  ideal  body  which  obeys  Hooke's  law  for  all  stresses, 
and  which  returns  precisely  to  its  original  condition  when  the 
stress  is  removed,  is  called  a  perfectly  elastic  body.  No  real 
bodies  are  perfectly  elastic  with  respect  to  both  of  the  funda- 
mental types  of  strain.  Liquids  and  gases,  possibly  also  solids, 
are  perfectly  elastic  with  respect  to  compression.  No  known 
body  is  perfectly  elastic  with  respect  to  shear. 

The  behavior  of  elastic  solids  under  the  action  of  forces,  or 
which  have  been  subjected  to  forces  for  a  time  and  then  re- 
lieved of  them,  depends  in  a  curious  and  very  complicated  way 
upon  the  time  during  which  the  force  acts  and  upon  the  time 
which  elapses  after  the  force  is  removed.  The  phenomena 
exhibited  by  a  body  in  these  circumstances,  and  in  others  gen- 


86  ELASTICITY. 

erally  similar  to  them,  are  ascribed  to  what  is  called  elastic 
fatigue. 

65.  Solids  and  Fluids. — Substances  differ  characteristically 
in  the  way  in  which  they  behave  under  stress.  In  particular, 
we  find  by  experiment  that  all  known  bodies  may  be  grouped 
in  one  of  two  classes,  according  to  the  way  in  which  they  be- 
have under  shearing  stress.  The  bodies  in  these  two  classes 
are  called  solids  and  fluids.  A  solid  is  a  body  which  will  offer 
a  permanent  resistance  to  a  shearing  stress;  that  is,  it  pos- 
sesses a  rigidity  which  depends  upon  permanent  and  conserva- 
tive forces  acting  betM-een  its  parts.  The  fluid,  on  the  other 
hand,  will  not  offer  a  permanent  resistance  to  a  shearing 
stress.  If  the  shearing  stress,  no  matter  how  small  it  is,  is 
only  applied  long  enough,  the  fluid  will  yield  to  it,  will  undergo 
a  continual  and  increasing  deformation,  and  will  not  tend  to 
recover  its  original  shape  when  the  stress  is  removed.  That 
is,  the  fluid  possesses  no  true  rigidity. 

It  is  not  to  be  understood  that  every  fluid  will  yield 
instantly  and  completely  to  a  shearing  stress.  All  fluids,  even 
those  whose  parts  move  easiest  among  themselves,  exhibit 
viscosity  or  internal  friction.  In  many  fluids  the  viscosity  ia 
exceedingly  great.  When  a  shearing  stress  is  applied  to  a 
fluid,  the  rate  at  which  the  fluid  yields  to  it  depends  upon  its 
viscosity,  and  the  time  taken  to  effect  a  perceptible  deforma- 
tion may  be  very  great. 


CAPILLARITY  87 


CAPILLARITY. 

66.  Capillarity. — If  we  dip  one  end  of  a  glass  tube  in  water 
and  examine  the  water's  surface  around  it  and  in  it,  we  find 
that  it  is  not  everywhere  level,  as  our  hydrostatic  theory  as- 
serts that  it  should  be.     Around  the  walls  of  the  tube,  both 
without  and  within,  the  water  rises  above  the  general  level. 
If  the  diameter  of  the  tube  is  small,  the  whole  free  surface 
within  the  tube  rises  above  the  general  level,  so  that  a  column 
of  water  is  lifted  by  it.    This  phenomenon  is  said  to  have  been 
studied  first  by  Leonardo  da  Vinci.     The  tubes  in  which  it 
appears  conspicuously  have  a  very  fine,  or  capillary  bore.    The 
general  subject  which  deals  with  this  phenomenon  and  with 
many  others  essentially  like  it,  and  due  to  the  same  general 
cause,  is  therefore  named  capillarity. 

67.  Surface   Tension. — The   general    laws    of   hydrostatics 
depend  upon  the  principle  that  a  liquid  subject  to  the  attrac- 
tion of  gravity  will  be  in  equilibrium  only  when  its  configura- 
tion is  such  that  the  action  of  gravity  on  it  introduces  no 
shearing  stresses.     Now  gravity  is  not  the  only  force  which 
acts  on  the  liquid.     Its  parts  also  exert  forces  of  cohesion  on 
each  other,  and  true  equilibrium  will  not  be  reached  until  the 
liquid  assumes  such  a  position  that  these  forces  of  cohesion  so 
act,  together  with  the  weights  of  the  parts  of  the  liquid,  as  to 
introduce    no    shearing    stresses.      We    can    explain    all    the 
phenomena    which    are    treated    under    capillarity    by    taking 
these  forces  of  cohesion  into  account.     It  is  not  necessary  for 
us  to  know,  and  in  fact  we  do  not  know,  the  way  in  which  the 
cohesion  depends  upon  the  masses  of  the  interacting  parts  and 
the  distances  between  them.     It  is  necessary,  however,  to  as- 
sume this  much,  that  the  force  of  cohesion  exerted  by  a  small 
part  or  element  of  the  body  only  acts  on  those  elements  which 
are  in  its  immediate  neighborhood.     That  is,  we  assume,  as 
the  law  of  the  force  of  cohesion  between  elements,  that  the 
force  between  contiguous  elements  is  very  great,  and  dimin- 


88  CAPILLARITY. 

ishes  very  rapidly  when  they  are  separated,  so  as  to  become 
imperceptible,  even  when  the  distance  between  them  is  still 
very  small.  This  general  law  of  cohesive  forces  is  illustrated 
by  the  behavior  of  an  iron  bar  when  we  break  it.  It  requires 
a  very  great  force  to  break  it,  but  after  it  is  broken,  even 
though  the  two  surfaces  at  the  break  are  fitted  together  again 
with  the  utmost  nicety,  the  two  parts  can  be  separated  with 
no  perceptible  effort. 

Because  of  the  short  distance  within  which  an  element  of 
the  liquid  acts  on  its  neighbors,  those  elements  which  lie 
below  the  surface  by  a  depth  equal  to  this  distance,  which  we 
call  the  range  of  action,  are  in  equilibrium  under  the  action 
of  their  neighboring  elements.  It  is  only  those  elements  which 
lie  in  or  very  near  to  the  surface  which  are  attracted  unequally 
in  different  directions.  These  unequal  attractions,  acting  on 
the  elements  of  the  surface  film,  will  produce  a  peculiar  con- 
dition in  it.  This  will  be  the  same  for  all  parts  of  the  surface, 
owing  to  the  minuteness  of  the  range  of  action,  so  long  as  the 
radius  of  curvature  of  the  surface  is  not  very  small,  that  is, 
is  not  of  the  same  order  of  magnitude  as  the  range  of  action. 
Thomas  Young  suggested  that  the  special  action  of  the  co- 
hesive forces  in  the  surface  film  may  be  represented  by  sup- 
posing the  film  to  be  under  tension,  similar  in  general  to  that 
in  a  stretched  membrane.  This  tension  should  be  the  same,  in 
the  case  of  any  given  liquid,  for  all  parts  of  its  surface,  it 
is  called  the  surface  tension,  and  its  numerical  value,  when 
determined  for  the  surface  of  separation  between  any  two 
bodies,  is  a  characteristic  number  for  those  bodies.  The 
position  of  the  liquid  column  in  the  capillary  tube,  or  any  of 
the  other  phenomena  ascribed  to  capillary  action,  are,  on  this 
view,  due  to  equilibrium  between  the  weights  of  the  parts  of 
the  liquid  and  the  forces  due  to  the  tensions  acting  in  curved 
portions  of  the  surface  film. 

Another  useful  concept  for  the  study  of  capillary  phenom- 
ena was  introduced  by  Gauss.  It  is  now  called  surface  energy. 
It  is  plain  from  the  consideration  of  the  action  of  the  cohesive 
forces,  that  when  the  surface  of  a  liquid  mass  is  enlarged,  the 


CAPILLARITY.  89 

potential  energy  of  the  liquid  is  increased  on  that  account. 
For,  the  surface  can  only  be  enlarged  by  the  passage  of  ele- 
ments of  the  liquid  from  the  interior  mass  into  the  surface 
film.  Each  of  the  elements  in  the  film  is  drawn  inward  toward 
the  interior  by  a  force  of  cohesion,  and  hence  negative  work  is 
done  on  all  the  elements  which  pass  out  from  the  interior  into 
the  surface  film.  The  negative  work  thus  done  is  equivalent 
to  an  increase  in  the  potential  energy  of  the  liquid.  The  sur- 
face, therefore,  possesses  an  energy  peculiar  to  itself,  propor- 
tional to  the  extent  of  surface  and  characteristic  of  the  bodies, 
separated  by  the  surface.  For  a  given  surface  its  numerical 
value  is  the  same  as  that  of  the  surface  tension  for  the  same 
surface. 

By  the  aid  of  this  concept  of  surface  energy  all  the  phenom- 
ena of  capillarity  may  be  explained  as  illustrations  of  the 
general  principle  that  the  potential  energy  of  a  system  of 
bodies  tends  to  become  the  least  possible. 

68.  Plateau's  Experiments. — If  a  limited  portion  of  liquid 
is  so  situated  that  its  equilibrium  does  not  depend  upon  its 
weight,  the  cohesive  forces  will  act  alone  to  determine  the 
form  which  the  liquid  will  assume  when  in  equilibrium.  In 
order  to  examine  the  behavior  of  a  liquid  in  these  circum- 
stances, Plateau  mixed  a  quantity  of  alcohol  and  water,  ad- 
justing the  proportions  of  the  two  ingredients  until  the  dens?- 
ity  of  the  mixture  was  the  same  as  that  of  olive  oil.  When 
masses  of  olive  oil  were  introduced  into  this  mixture,  they 
had  no  tendency  to  rise  or  sink,  their  weights  therefore  did 
not  effect  their  equilibrium,  and  the  cohesive  forces  were  free 
to  act  alone.  Each  of  the  masses  of  oil  assumed  a  spherical 
form,  provided  they  were  entirely  free  in  the  supporting  mix- 
ture. This  result  is  consistent  with  our  general  law  of  equi- 
librium; for,  equilibrium  can  exist  in  a  mass  only  when  the 
pressure  is  the  same  in  it  everywhere,  and  in  order  that  the 
pressure  should  be  the  same,  the  curvature  of  the  surface  film 
should  be  the  same  everywhere.  Or,  looking  at  the  same  mat- 
ter from  the  point  of  view  of  surface  energy,  the  form  which 


90  CAPILLARITY. 

the  mass  will  assume  will  be  that  having  the  leasl  potential 
energy  and  therefore  the  least  surface. 

If  the  mass  of  oil  is  not  free,  but  suspended  on  a  wire  frame 
work,  its  surface  will  assume  different  forms.  Whatever  be 
the  form  of  these  surfaces,  they  must  be  such  that  tensions  in 
them  of  equal  value  will  give  rise  to  equal  pressure  within  the 
liquid  mass.  Analysis  has  shown  that  this  condition  is 
attained  when  the  curvature  of  the  surfaces  is  such  that  the 
sum  of  the  reciprocals  of  the  principal  radii  of  curvature  is 
•the  same  at  every  point  on  them. 

Minimal  or  ruled  surfaces  nr«  represented  by  films  of 
soapy  water.  The  simplest  surface  of  this  kind,  the  sphere,  is 
illustrated  by  the  ordinary  soap  bubble.  The  weight  of  the 
film  is  so  small  a  force,  in  comparison  with  the  forces  intro- 
duced by  the  surface  tension,  that  the  figure  due  to  the  sur- 
face tension  is  scarcely  distorted  at  all  by  the  weight  of  the 
film.  If  a  wire  framework  is  dipped  in  the  soap  solution  and 
taken  out  again,  films  of  water  will  usually  adhere  to  it. 
When  a  part  of  the  frame  work  lies  in  a  plane,  the  film 
attached  to  it  is  also  plane.  If  it  is  distorted,  the  film  exhibits 
the  peculiarities  of  the  minimal  surface.  When  three  films 
meet,  they  always  meet  along  a  straight  line,  and  make  equal 
angles  with  each  other.  Four  films  or  more  meet  in  a  point. 
The  angles  between  the  films  which  meet  in  this  manner  are 
determined  by  the  general  condition  that  the  surface  of  the 
films  is  a  minimum. 

69.  Rise  of  Liquids  in  Tubes. — When  a  thoroughly  clean 
glass  rod  is  dipped  in  water,  the  water  rises  around  its  sides 
and  wets  it.  If  the  rod,  on  the  other  hand,  is  dipped  in  mer- 
cury, the  mercury  is  depressed  around  it  and  does  not  wet  it. 
We  ascribe  this  different  behavior  in  these  typical  cases  to  the 
different  relative  values  of  the  cohesive  forces  of  water  and 
of  mercury  and  the  different  values  of  the  forces  with  which 
the  glass  attracts  those  substances.  To  describe  the  wetting 
of  the  rod  by  the  use  of  the  idea  of  surface  tension,  we  may 
say  that  the  tension  in  the  surface  separating  the  glass  and 
air  is  greater  than  the  sum  of  the  tensions  in  the  surfaces 


CAPILLARITY.  91 

separating  water  and  air,  and  glass  and  water,  so  that  the 
edge  of  the  liquid,  where  it  meets  the  glass,  is  pulled  upward 
by  the  strongest  tension.  It  is  probably  true  that  for  very 
clean  glass  the  result  of  this  action  is  to  cover  the  whole  of 
the  glass  with  a  very  thin  film  of  water. 

If  a  narrow  tube  of  circular  bore  is  immersed  in  a  liquid 
which  wets  it,  the  liquid  will  rise  in  the  tube.  The  summit  of 
the  column  of  liquid  is  terminated  by  a  cup-shaped  or  almost 
hemispherical  surface,  called  the  meniscus.  The  height  of  the 
column,  which  is  in  equilibrium  between  its  own  weight  and 
the  upward  pull  on  it,  due  to  the  surface  tension,  is  inversely 
proportional  to  the  radius  of  the  tube.  This  law  is  known  as 
Jurin's  law.  It  can  be  seen  at  once  without  formal  demon- 
stration that  the  theory  of  surface  tension  leads  to  this  law. 
For,  the  surface  tension  which  lifts  the  column  acts  on  every 
point  of  the  circumference  of  the  upper  surface,  or  along  a 
line  which  is  proportional  to  the  radius  of  the  tube.  The  col- 
umn which  it  lifts  has  a  weight  proportional  to  the  height  and 
to  the  area  of  its  cross  section;  that  is,  proportional  to  its 
height  and  to  the  square  of  the  radius  of  the  tube.  Since  the 
upward  pull  due  to  the  surface  tension  is  equal  to  the  weight, 
the  height  of  the  column  is  proportional  to  the  surface  tension 
and  to  the  radius  and  is  inversely  proportional  to  the  area  of 
its  cross  section,  or  to  the  square  of  the  radius.  Therefore 
finally  the  height  of  the  column  is  inversely  as  the  radius  of 
the  tube. 

If  a  small  portion  of  liquid  is  contained  in  a  long  capillary 
tube  whose  bore  is  not  cylindrical,  but  conical,  it  will  move 
toward  the  small  end  of  the  tube,  because  it  is  the  end  of  the 
column  whose  radius  is  less  upon  which  the  surface  tension 
acts  more  advantageously.  If  the  column  of  liquid  is  in  a 
cylindrical  tube,  it  will  move  toward  the  end  whose  surface 
tension  is  greater.  We  may  diminish  the  surface  tension  of 
one  end  of  the  column  by  raising  its  temperature,  or  by  cov- 
ering it  with  a  film  of  another  liquid  of  lower  surface'  tension. 

70.  Bubbles  and  Drops.— A  bubble  is  a  mass  of  air  sur- 
rounded by  a  liquid.  The  shape  of  the  liquid  surface  is  de- 


92  CAPILLARITY. 

termined  by  the  condition  that  the  pressure  at  every  point, 
due  to  the  surrounding  liquid,  in  addition  to  the  pressure  due 
to  the  surface  tension,  shall  be  equal  to  the  pressure  of  the 
air  in  the  bubble.  For  bubbles  formed  within  films  in  the  air, 
substantially  the  same  conditions  of  equilibrium  hold,  except 
that  the  weight  of  the  film  is  almost  negligible,  and  equi- 
librium exists  when  the  external  atmospheric  pressure,  in  addi- 
tion to  the  pressure  due  to  the  surface  tension,  is  equal  to  the 
pressure  of  the  air  within  the  bubble.  The  bubble  of  typical 
form  is  one  made  by  introducing  air  under  a  flat  glass  plate 
held  horizontally  in  the  liquid.  The  plate  keeps  it  from  rising, 
and  it  assumes  a  condition  of  equilibrium  determined  by  the 
balance  of  pressures  already  described.  When  the  bubble  is 
very  minute,  its  form  is  approximately  spherical.  As  more 
air  is  introduced  it  widens  out  and  increases  in  height.  Its 
height  depends  upon  the  surface  tension  of  the  liquid,  and 
tends  toward  a  limiting  value,  as  the  volume  of  the  bubble 
increases,  of  not  more  than  a  few  millimetres  in  any  known 
case.  Any  horizontal  cross-section  of  the  bubble  is  circular. 
Its  under  surface  is  fiat,  when  the  bubble  exceeds  a  certain 
size,  and  over  that  flat  surface  the  pressure  of  the  air  within 
the  bubble  is  equal  to  the  pressure  of  the  liquid  column  whose 
height  is  the  height  of  the  bubble.  For  points  on  the  side  of 
the  bubble,  at  which  the  surface  is  curved,  an  additional 
pressure  exists  due  to  the  surface  tension. 

A  drop  is  simply  an  inverted  bubble,  a  mass  of  liquid  rest- 
ing on  a  support  and  surrounded  by  air.  A  drop  of  mercury 
on  a  flat  surface  is  a  typical  form.  With  evident  modifications, 
what  has  been  said  about  the  bubble  applies  equally  well  to 
the  drop.  Drops  of  water,  or  of  other  liquids  which  wet  solids, 
are  usually  limited  in  respect  of  the  area  of  the  base  on  which 
they  stand  either  by  impurities  on  the  surface  of  the  solid, 
which  prevent  its  being  wetted,  or  by  abrupt  changes  in  the 
shape  of  the  solid.  The  theory  of  the  equilibrium  of  such 
drops  is  essentially  the  same  as  that  already  stated,  but  special 
cases  sometimes  need  careful  study  in  order  to  perceive  that 


CAPILLARITY. 

they  are  not  inconsistent  with  the  general  principles  already 
laid  down. 

71.  Surface  Viscosity. — The  parts  of  some  liquids  move 
with  much  less  freedom  among  themselves  than  those  of  others. 
The  forces  with  which  they  resist  the  force  which  moves  them 
are  non-conservative  forces  and  do  not  tend  to  restore  the  parts 
which  have  been  displaced  to  their  original  position.  They 
are  ascribed  to  the  mutual  forces  between  the  parts,  coupled 
with  the  mobility  of  those  parts.  The  cause  of  this  action  is 
called  internal  friction  or  viscosity. 

In  many  liquids,  probably  in  all  of  them,  the  viscosity  In 
the  surface  film  is  different  from  that  in  the  general  mass.  Its 
value  is  often  very  much  greater  than  that  in  the  mass.  The 
value  of  a  liquid,  as  a  means  of  illustrating  the  principles  of 
capillarity,  often  depends  very  much  on  the  fact  that  it 
possesses  a  considerable  surface  viscosity.  For  example,  the 
surface  tension  of  water  is  greater  than  that  of  any  other 
ordinary  liquid,  and  pure  water  would  therefore  seem  to  be 
the  best  liquid  that  could  be  used,  in  experimenting  on  capil- 
larity. On  the  other  hand  the  surface  viscosity  of  water  is  not 
great,  and  the  parts  of  a  film  of  pure  water  will  move  so 
freely  under  their  own  weight  that  the  film  very  soon  becomes 
somewhere  thin  enough  to  break.  The  addition  of  soap  to  the 
water,  so  as  to  make  a  strong  soap  solution,  diminishes  the 
surface  tension,  but  very  considerably  increases  the  surface 
viscosity,  so  that  a  film  of  soap  solution,  whose  parts  do  not 
move  freely,  will  remain  unbroken  for  so  long  a  time  that  its 
form  and  peculiarities  may  be  studied. 


GASES. 

72.  Mature  of  a  Gas. — The  experiment  of  Torricelli  with 
the  barometer  showed  that  it  was  possible  to  obtain  a  region 
in  space  that  was  to  all  appearance  void  or  empty  of  matter. 
This  region  above  the  mercury  of  the  barometric  column. was 
called  the  Torricellian  vacuum. 

It  occurred  to  von  Guericke  that  it  might  be  possible  to 
produce  a  similar  vacuum  by  withdrawing  air  from  a  closed 
vessel  by  means  of  a  pump.  After  several  unsuccessful  at- 
tempts, von  Guericke  at  last  succeeded  in  doing  this,  if  not  so 
completely  as  in  the  Torricellian  vacuum,  at  least  sufficiently 
well  to  enable  him  to  study  the  behavior  of  bodies  in  a  place 
void  of  air/  He  showed  that  if  a  partially  collapsed  bladder, 
tied  tightly  at  the  neck,  so  that  the  air  in  it  could  not  escape, 
was  placed  in  the  vessel  or  receiver  of  his  air  pump,  and  the 
air  removed  from  around  it,  the  air  in  it  would  swell  out  until 
the  bladder  seemed  full.  Thus  the  air  in  the  bladder  appeared 
to  expand  in  all  directions.  Its  behavior  in  this  respect  is  in 
marked  contrast  to  that  of  a  solid  or  of  a  liquid  placed  in  the 
receiver,  neither  of  which  will  expand  in  all  directions  when 
the  air  is  removed.  A  body  which  exhibits  this  property  of 
expansion  in  all  directions,  or,  as  it  is  sufficient  to  say,  of  ex- 
erting pressure  in  all  directions  on  the  walls  of  any  closed 
vessel  containing  it,  is  called  a  gas. 

73.  The  Air  as  a  Fluid. — A  limited   portion   of   air   has 
weight.     This  fact  was  recognized  by  Galileo,  who  attempted 
to  prove  it,  without  complete  success,  by  first  weighing  a  glass 
vessel  full  of  air,  and  weighing  it  again  after  some  of  the  air 
had  been  expelled  by  heating  it.    Aristotle  failed  in  an  attempt 
to  prove  the  same  thing.    He  weighed  a  bladder  full  of  air,  and 
then  after  forcing  the  air  out  of  the  bladder,  weighed  it  again. 
Of  course  he  found  no  difference  between  the  two  weights,  be- 
cause the  air  which  was  outside  the  bladder  in  the  second  trial 
took  the  place  of,  and  weighed  as  much  as  the  air  which  was 


OASKS.  95 

within  it  in  the  first  trial.  The  experiment  cannot  succeed 
unless  the  walls  of  the  vessel  are  rigid,  and  unless  the  air  can 
be  removed  from  its  interior.  By  employing  an  experiment  of 
this  sort,  von  Guericke  established  the  fact  that  air  has  weight. 

This  being  so,  and  air  being  a  fluid,  the  pressure  relations 
of  which  are  the  same  as  those  of  a  liquid,  it  follows  that,  with 
certain  modifications,  hydrostatic  theorems  apply  to  air  as 
well  as  to  liquids.  In  particular  Archimedes'  principle  appliss 
to  botli  classes  of  bodies.  A  body  weighs  less  in  air  than  it 
would  weigh  in  vacuum,  by  an  amount  equal  to  the  weight  of 
the  air  which  it  displaces.  In  very  exact  weighing,  it  is  there- 
fore necessary  to  take  account  of  the  volumes  of  the  bodies 
which  are  weighed,  and  of  the  weights  used,  in  order  to  obtain 
the  weights  of  the  bodies  in  vacuum. 

In  1755  Black  discovered  the  gas  called  carbon  di-oxide. 
Soon  afterward  hydrogen  gas  was  discovered.  This  was  soon 
followed  by  the  discovery  of  many  other  gases,  and  by  the 
recognition  of  the  fact  that  the  vapors  formed  by  the  evapo- 
ration of  volatile  liquids  were  in  most  essential  respects  sim- 
ilar to  gases.  Thus  to  the  two  classes  of  bodies  which  had 
long  been  recognized,  solids  and  liquids,  there  was  added  a 
third  class,  gases.  For  our  present  purposes,  we  may  define  a 
solid  as  a  body  which  will  retain  its  form  and  volume  un- 
changed, under  the  action  of  its  own  internal  forces.  A  liquid 
is  a  body  which  will  retain  its  volume  unchanged,  under  the 
action  of  its  own  internal  forces,  but  which  yields  to  shearing 
stress,  so  that  it  cannot  retain  a  definite  form,  except  when 
placed  in  some  receptacle.  The  walls  of  this  receptacle  exert 
pressure  on  it  of  such  a  sort  as  to  annul  the  shearing  stresses 
in  it.  A  gas  is  a  body  whose  internal  forces  do  not  constrain 
it  to  assume  any  definite  volume  or  form.  It  expands  in  all 
directions,  so  that  it  cannot  be  confined  as  a  liquid  can,  in  a 
vessel  open  at  the  top.  When  confined  in  a  closed  vessel,  it 
exerts  pressure  upon  every  part  of  the  surface  which  en- 
velops it. 

74.  Boyle's  Law. — The  experiments  of  von  Guericke  with 
the  air  pump  were  repeated  and  amplified  by  Robert  Boyle. 


His  attention  was  thus  attracted  tu  the  relation  between  the 
volume  of  a  limited  portion  of  air  and  the  pressure  upon  it, 
and  it  occurred  to  him  that  it  might  be  worth  while  to  obtain 
a  set  of  values  of  the  volumes  and  of  the  corresponding  pres- 
sures. He  did  this  by  isolating  a  small  quantity  of  air  in  the 
short  limb  of  a  U-shaped  tube  by  means  of  a  quantity  of  mer- 
cury in  the  bend  of  the  tube.  When  this  mercury  was  so 
adjusted  that  its  ends  stood  at  the  same  level  in  both  limbs  of 
the  tube,  the  pressure  on  the  enclosed  air  was  that  of  the 
atmosphere.  When  more  mercury  was  poured  into  the  long 
limb  of  the  tube  the  surface  in  the  short  limb  rose  and  the  air 
was  compressed.  The  pressure  on  it  was  that  of  the  atmo- 
sphere, increased  by  the  pressure  of  a  mercury  column  whose 
neight  was  the  difference  of  level  between  the  surfaces  of  the 
mercury  in  the  two  limbs  of  the  tube.  By  means  of  a  number 
of  measurements  of  this  sort  Boyle  showed  that  the  volume 
of  the  air  in  the  tube  varied  inversely  with  the  pressure  upon 
it.  To  illustrate  this  statement  we  shall  consider  the  air  con- 
fined in  the  tube  under  atmospheric  pressure.  We  may  take 
the  volume  of  the  air  in  this  condition  as  unit  volume,  and 
the  pressure  of  one  atmosphere  as  unit  pressure.  If  mercury 
is  then  poured  in  until  the  pressure  becomes  two  atmospheres, 
the  volume  of  the  air  is  reduced  to  one-half  its  original  vol- 
ume. If  the  pressure  is  made  three  atmospheres,  the  volume 
of  the  air  is  one-third  its  original  volume,  and  so  on.  For 
sach  of  these  cases  the  product  of  the  volume  and  the  corre- 
sponding pressure  is  equal  to  1.  This  product  might  have  had 
any  numerical  value  we  pleased  to  give  it,  depending  upon  our 
choice  of  the  volume  taken  as  unit  volume  and  the  pressure 
taken  as  unit  pressure.  But  whatever  that  value  was,  the 
experiments  show  that  the  product  of  the  volume  and  the 
corresponding  pressure  will  always  be  the  same.  We  may, 
therefore,  express  Boyle's  law  in  another  and  more  convenient 
form,  by  saying  that  the  product  of  the  volume  of  a  given 
mass  of  air  and  the  pressure  upon  it  is  constant. 

Boyle's  law  was  subsequently  shown  to  hold  not  only  for 
air,  but  also  for  all  gases.    If  equal  volumes  of  different  gases 


are  taken  under  equal  pressures,  the  volumes  of  all  of  them 
will  change,  when  the  pressures  are  changed,  according  to  the 
same  law.  The  law  holds  not  only  for  the  case  in  which  the 
pressure  is  continually  increased,  but  also  for  that  in  which 
the  pressure  is  decreased.  Gases,  therefore,  differ  from  solids 
and  liquids  in  that  they  all  possess  the  same  volume  elasticity. 

75.  Cray-Lussac's  Law. — Gases  expand  when  heated.     This 
fact  was  known  to  Galileo,  who  used  it  in  the  construction  of 
the  first  thermometer.     Owing  to  various  causes,  the  law  of 
this  expansion  was  not  discovered  until  many  years  after  Gali- 
leo's time.    In  1809  Gay-Lussac  showed  that  all  gases  expand 
at  the  same  rate  as  their  temperatures  rise.    That  is,  when  the 
temperatures  of  different  gases  are  raised  from  that  of  the 
melting  point   of  ice  to  that   of  the  boiling  point   of  water, 
their   volumes   all  increase   by   the   same   proportionate   amount. 
Gases   differ   from  solids  and  liquids  in  that  they  all  possess  the 
same  coefficient  of  expansion  with  rise  of  temperature. 

This  law  was  discovered,  though  not  published,  by  the 
chemist  Charles,  the  inventor  of  the  hydrogen  balloon.  It  is 
therefore  sometimes  known  as  Charles'  law. 

76.  Law  of  Combining  Volumes.    Avogadro's  Law. — As  the 
result    of   his    experiments    on    the    chemical    combination    of 
gases,  Gay-Lussac  showed  that,  in  the  case  of  complete  chem- 
ical combination,  the  volumes  of  the  gases  which  combine  and 
the  volume  of  the  resultant  product,  if  it  also  is  a  gas,  are  in 
a  simple  numerical  relation  to  one  another.    For  example,  two 
volumes  of  hydrogen  will  combine  completely  with  one  volume 
of  oxygen,  and  as  the  result  of  that  combination,  two  volumes 
of  water  vapor  are  obtained.    The  volumes  are  of  course  meas- 
iired  at  the  same  pressure  and  temperature.     The  law  thus 
stated  is  called  the  law  of  combining  volumes. 

An  explanation  of  this  law  was  given  in  1811  by  Avogadro, 
and  in  1811  by  Ampere. 

It  had  for  some  time  been  believed  that  bodies  did  not  con- 
sist of  matter  continuously  distributed,  but  that  the  matter 
of  a  body  was  collected  in  separate  particles.  Newton 
described  these  particles  in  the  following  words:  "It  seems 


probable  to  me  that  God  in  the  beginning  formed  matter  in 
solid,  massy,  hard,  impenetrable,  movable  particles,  of  such 
sizes  and  figures  and  with  such  other  properties  and  in  such 
proportion  to  space  as  most  conduced  to  the  end  for  which 
He  formed  them;  and  that  these  primitive  particles,  being 
solids,  are  incomparably  harder  than  any  porous  bodies  com- 
pounded of  them,  even  so  very  hard  as  never  to  wear  or  break 
in  pieces;  no  ordinary  power  being  able  to  divide  what  God 
Himself  made  one  in  the  first  creation.  ...  It  seems  to 
me,  farther,  that  these  particles  have  not  only  a  vis  inertiae, 
accompanied  with  such  passive  laws  of  motion  as  naturally 
result  from  that  force,  but  also  that  they  are  moved  by  cer- 
tain active  principles."  The  particles  of  gases  were  supposed 
to  be  of  this  sort. 

Avogadro  perceived  that  the  law  of  combining  volumes 
could  not  be  explained  by  assuming  the  particles  of  the  gas 
to  be  such  particles  as  these.  He  therefore  assumed  that  the 
particles  which  compose  the  gas,  and  which  give  to  it  its  char- 
acteristic physical  properties,  are  combinations  of  two  or  more 
elementary  particles.  When  these  elementary  particles  are  of 
the  same  kind,  the  substance  made  up  of  them  is  a  chemical 
element.  When  they  are  of  different  kinds,  the  substance  is 
a  compound.  Chemical  combination  then  involves  a  breaking 
up  of  the  groups  of  each  of  the  elements,  and  a  combination 
of  the  particles  composing  these  groups  with  each  other  to 
form  new  groups.  The  elementary  particles  are  called  atoms, 
the  groups  composed  of  them,  molecules.  These  names  were 
first  given  by  Ampere. 

Avogadro  announced  the  law  that  equal  volumes  of  differ- 
ent gases,  under  the  same  conditions  of  pressure  and  tempera- 
/  ture,  contain  equal  numbers  of  molecules.  With  this  law  as 
a  foundation  he  was  able  to  explain  the  law  of  combining 
volumes. 

No  such  law  as  this  can  be  shown  to  hold  for  solids  and 
liquids.  Gases,  therefore,  differ  from  solids  and  liquids  in  that 
equal  volumes  of  them,  under  similar  conditions,  contain  equal 
numbers  of  molecules. 


GASES.  99 

77.  Molecular  Theory  of  Matter. — This  theory  of  the  con- 
stitution of  bodies,   which  has  been  forced  upon  us  by  the 
study  of  gases,  affords  a  complete  explanation  of  all  the  chem- 
ical reactions  between  bodies.     We  therefore  conclude  that  a 
homogeneous  body  is  composed  of  similar  molecules,  and  that 
these  molecules  are  composed  of  atoms.    In  a  few  cases  a  mole- 
cule contains  only  one  atom,  or  the  molecule  and  the  atom  are 
identical.     Broadly  speaking,  the  science  of  chemistry  is  con- 
cerned with  the  study  of  the  various  elementary  atoms  and 
of  their  possible  combinations.     The  science  of  physics  deals 
with  the  properties  of  bodies  in  so  far  as  they  depend  upon 
the  peculiarities  of  their  molecules. 

In  most  physical  operations,  the  molecules  of  the  bodies 
which  are  studied  are  unchanged  by  those  operations.  In 
chemical  operations,  the  molecules  break  up,  and  new  mole- 
cules of  other  sorts  are  formed.  The  elementary  atoms  are 
not  appreciably  changed  in  any  ordinary  chemical  operations, 
and,  until  very  recently,  they  have  always  been  assumed  to 
be  indestructible.  But  the  study  of  radioactive  bodies  has 
shown  that  it  is  highly  probable  that,  at  least  in  certain 
cases,  the  atom  is  really  a  composite  body,  and  that  it  may 
change  its  character  by  the  loss  of  some  of  its  parts.  We  may, 
for  the  present,  assume  the  atom  to  be  indestructible. 

78.  The  Kinetic  Theory  of  Gases. — The  properties  possessed 
in  common  by  all  gases  indicate  a  common  cause  to  which 
those  properties  may  be  ascribed.     As  far  back  as  the  year 
1738,   an  attempt  was  made   by  Daniel   Bernoulli  to   explain 
them  as  the  result  of  the  purely  mechanical  impact  of  the 
particles  or  molecules  of  the  gas.    Bernoulli  assumed  that  the 
molecules  of  a  gas  are  in  constant  motion,  and  that  they  col- 
lide with  each  other  and  with  the  walls  of  the  vessel  which 
contains  them.     He  also  assumed  that  their  number  in  any 
ordinary  volume  is  enormously  great.    It  is  evident  that  the 
effect  of  their  collisions  with  each  other  will  be  to  alter  the 
velocities  of  the  individual  molecules,  so  that,  even  if  they 
were  equal  at  any  time,  they  would  not  long  remain  the  same. 
For  the  purposes  of  elementary  calculation,  we  may  assume 


100  GASES. 

that  the  effect  which  they  produce  by  their  impacts  against 
the  walls  of  the  vessel  which  contains  them  is  the  same  as 
that  which  they  would  produce  if  they  all  had  a  common  veloc- 
ity. The  theory  asserts  that  the  pressure  of  the  gas  on  the 
walls  of  the  containing  vessel  is  due  to  the  impacts  of  the 
molecules.  The  problem  then  is  to  determine,  on  these  sup- 
positions, how  the  pressure  on  the  walls  wrill  depend  upon  the 
volume  of  the  vessel  and  the  velocity  of  the  molecules.  Ber- 
noulli solved  this  problem  in  the  following  way: 

The  gas  is  supposed  to  be  contained  in  a  cylindrical  vessel 
and  confined  within  it  by  a  piston,  which  at  first  stands  at  unit 
distance  from  the  base  of  the  cylinder.  If  the  piston  is  then 
pushed  down  until  its  distance  from  the  base  is  s,  the  volume 
occupied  by  the  gas  may  be  represented  by  s,  if  its  original 
volume  is  represented  by  1.  It  is  assumed  that  the  common 
velocity  of  the  molecules  is  not  changed  by  this  operation. 
Now  there  are  two  reasons  why  the  pressure  of  the  gas  should 
be  greater,  when  its  volume  is  thus  reduced,  than  it  was  be 
fore:  first,  the  molecules  have  been  crowded  together  into  a 
smaller  volume,  and  consequently  there  will  be  more  mole- 
cules lying  near  the  walls;  secondly,  the  average  volume  oc- 
cupied by  any  one  molecule  is  diminished,  and  consequently 
the  molecule  will  strike  the  side  of  this  volume  more  fre- 
quently. If  we  consider  the  first  cause  assigned,  it  appears 
that  the  number  of  molecules  which  will  lie  contiguous  to  unit 
area  of  the  wall  will  be  inversely  proportional  to  the  square 
of  the  cube  roots  of  the  volumes  of  the  gas,  so  that  the  ratio 
of  the  pressures  in  the  two  conditions  of  the  gas,  so  far  is 

they  depend  on  the  first  cause,  will  be  given  hy  s% :  I.  If  we 
consider  the  second  cause  assigned,  it  appears  that  the  number 
of  impacts  of  any  one  molecule  against  the  walls  will  be  in- 
versely as  the  average  distance  between  the  particles,  or  in- 
versely as  the  cube  roots  of  the  volumes  of  the  gas,  so  that 
the  ratio  of  the  pressures  in  the  two  conditions  of  the  gas,  so 
far  as  it  depends  on  the  second  cause,  will  he  sriven  hy  s^:  1. 
When  we  consider  that  both  the  causes  assigned  act  to- 


GASES.  101 

gether,  we  conclude  that  the  ratio  of  the  pressures  in  the  two 

conditions  of  the  gas  is  given  by  (s*)  (s^):l,  or  by  s:l,  and 
hence  conclude,  as  the  result  of  our  original  hypothesis  as  to 
the  nature  of  a  gas,  and  the  way  in  which  it  exerts  pressure, 
that  when  the  velocity  of  the  molecules  of  a  gas  is  maintained 
constant,  its  pressure  varies  inversely  with  its  volume. 

To  show  the  way  in  which  the  pressure  depends  upon  the 
velocity  of  the  molecules,  when  the  volume  is  unchanged,  we 
notice  that  the  impulse  applied  by  each  molecule  to  the  wall, 
measured  by  its  change  in  velocity,  is  proportional  to  its 
velocity,  and  that  the  number  of  impulses  which  each  molecule 
exerts  upon  the  wall  in  unit  time  is  also  proportional  to  its 
velocity,  so  that  the  effect  due  to  any  one  molecule,  and  there- 
fore the  effect  due  to  their  combined  action,  is  proportional  to 
the  square  of  their  velocity.  If  we  assume  that  the  velocity 
of  the  molecules  is  increased  by  the  introduction  of  heat,  we 
may  explain  in  this  way  the  increase  of  the  pressure  which  a 
gas  exerts  when  its  temperature  rises,  and  by  the  use  of  Gay- 
Lussac's  law  we  may  obtain  a  simple  relation  between  the 
velocity  of  the  molecules  and  the  temperature  of  the  gas. 

The  success  of  the  kinetic  theory  in  explaining  the  behavior 
of  gases  has  been  so  complete,  and  its  influence  in  establishing 
the  general  kinetic  theory  of  matter  has  been  so  great,  that  it 
is  worth  while  to  discuss  this  question  more  formally,  so  as 
to  obtain  the  fundamental  equation  of  the  kinetic  theory  of 
gases.  We  shall  retain  the  same  suppositions  as  those  which 
have  been  made  already.  Let  us  suppose  that  n  molecules  of 
the  gas  are  contained  in  unit  volume,  and  that  this  volume  is 
a  cube.  The  number  of  molecules  contiguous  to  one  face  of 

this  unit  cube  will   then  be  n%.      The  average  volume  occupied 

bv  each  molecule  is  _,  and  the  edge  of  the  cube  containing  this 
n 

volume,  or  the  average  distance  through  which  a  molecule 
moves  in  passing  from  one  side  of  its  molecular  volume  to  the 

other,  is  ( ~\  .     A   molecule  which   lies  contiguous  to  the  sur- 
\»/ 


102  CASKS. 

face  will  pass  over  t.wice  this  distance,  with  tlie  average 
velocity  u.  in  the  time  which  elapses  between  two  successive 
impacts.  The  time  between  these  impacts  is  therefore  given  by 


\n'        The  number  of  impacts  which  the  tn<  lecule  will  make 
u 

in  unit  time  is  the  reciprocal  of  this,  or  _  — 

We  now  proceed  to  examine  the  effect  produced  on  the  wall 
by  the  impact  of  a  molecule.  To  do  this,  it  is  necessary  to 
make  the  supposition  that  the  average  energy  of  the  molecule 
is  not  changed  by  collision,  so  that  the  numerical  value  of  its 
velocity  after  collision  with  the  wall  is  the  same  as  it  was 
before  collision.  We  consider  a  molecule  moving  directly 
toward  the  wall.  Its  momentum  before  it  meets  the  wall  is 
mu,  and  its  momentum  after  it  leaves  the  wall  is  of  the  same 
numerical  value  and  in  the  opposite  direction.  The  total 
change  in  momentum,  which  measures  the  impulse  exerted  by 
the  wall  against  the  molecule,  is  therefore  2mu. 

The  molecules  are  moving  in  all  directions,  so  that  those 
which  lie  contiguous  to  the  wall  are  not  all  moving  per- 
pendicular to  it  at  once.  But  we  may  assume  that  one-third 
of  them  are  moving  in  this  way.  The  impulse  applied  to  unit 
surface,  in  the  small  time  t,  is  equal  to  the  impulse  due  to  a 
single  molecule,  multiplied  by  one-third  of  the  number  of 
molecules  which  lie  contiguous  to  the  unit  surface,  multiplied 
by  the  number  of  impacts  which  one  molecule  will  make  in 
unit  time,  multiplied  by  the  time  t.  This  impulse  is  counter- 
acted by  and  so  is  equal  to  the  average  force  exerted  by  the 
wall,  multiplied  by  the  same  time  t;  and  this  average  force 
acting  on  unit  surface  measures  the  pressure  exerted  by  the 
wall,  and  therefore  the  pressure  of  the  gas.  Collecting  the 

f  T 

results  of  the  previous  paragraphs,  we  obtain  pi  •=  2mu  -  '.  !!!*  .  t, 

3       *2 

or  p=$nmu*.    The  equation  thus  obtained  is  the  fundamental 
equation  of  the  kinetic  theory  of  gases. 


OA.SKS.  103 

The  number  of  molecules  represented  by  n  in  this  equation 
is  the  number  contained  in  unit  volume.  If  we  suppose  the 
amount  of  gas  to  remain  unchanged,  so  that  n  represents  the 
number  of  its  molecules  in  any  circumstances,  and  suppose  its 
volume  to  be  changed  from  unit  volume  to  the  volume  v,  the 

number   contained  in   unit   volume  will   be  ",   and   substituting 

v 

this  for  n  in  the  fundamental  equation  for  the  pressure,  we 
obtain  pv=$nmu*.  The  quantity  on  the  right  of  this  equation 
remains  a  constant,  if  the  common  velocity  of  the  molecules 
remains  constant.  Hence  we  conclude  that,  on  the  hypotheses 
assumed  at  the  outset,  the  product  of  the  pressure  and  volume 
of  the  gas  should  remain  constant,  if  the  velocity  of  its  mole- 
cules is  constant.  This  relation  is  that  which  was  obtained 
experimentally  by  Boyle. 

We  may  extend  this  demonstration  somewhat  by  making 
an  hypothesis  about  the  velocities  of  the  molecules,  which  more 
nearly  represents  their  true  velocities  than  the  one  which  we 
have  hitherto  used.  We  suppose  that  the  velocities  are  not 
the  same,  but  different.  The  number  of  molecules  is,  however, 
so  great  that  there  will  still  be  a  large  number.  «t  whose 
velocities  lie  very  near  a  common  value  MJ.  This  set  of  mole- 
cules will  exert  a  partial  pressure  pt  given  by  the  formula 
p^o  =  ^MjWietJ.  Similar  partial  pressures  will  be  exerted  by  other 
sets  of  molecules,  whose  velocities  lie  near  the  values  u2,  u3,  Ui, 

The  sum  of  all  these  partial  pressures  is  the 

pressure  of  the  gas.  If  we  represent  by  n  the  number  of  mole- 
cules of  the  gas,  by  u  a  certain  average  velocity,  by  «15  n2,  ns, 

the  numbers  of  molecules  in  the  different  sets 

which  have  been  considered,  we  may  write  an  equation  de- 
finding  the  average  velocity  as  follow?  :  mi2  =  n^  +  w,rt*  +  nzu\ 

+ The  average  velocity  thus  defined  is  called 

the  velocity  of  mean  square.  In  terms  of  this  velocity  we 
obtain  the  equation  pv=$nmnz  for  the  pressure  of  the  gas. 

As  has  already  been  remarked,  the  equation  just  obtained 
expresses  Boyle's  law.  As  will  be  seen  later,  when  taken  in 
connection  with  Gav-Lussac's  law,  it  affords  us  a  measure  of 


104  GASES. 

temperature  and  an  insight  into  the  nature  of  heat.  By  the 
aid  of  an  additional  theorem  first  given  by  Maxwell,  it  may  be 
made  to  demonstrate  Avogadro's  law  also.  Maxwell  proved 
that  when  two  gases  are  mixed  their  most  probable  condition, 
or  their  condition  of  final  stability,  is  that  in  which  the  mean 
kinetic  energies  of  the  molecules  of  both  gases  are  equal.  This 
being  so,  if  we  consider  equal  volumes  of  two  gases  under  the 
same  pressure  and  at  the  same  temperature,  so  that  they  con- 
form to  the  condition  of  being  in  mechanical  and  thermal 
equilibrium  with  each  other,  the  expression  \nmui  is  the  same 
for  each.  By  Maxwell's  law  the  factor  »iMa  is  also  the  same 
for  each,  so  that  the  factor  n  will  be  the  same  for  each.  Hence 
Avogadro's  law  follows  as  a  consequence  of  the  kinetic  theory. 
It  was  shown  by  Joule  that  we  may  calculate  the  average 
velocity  of  the  molecules  of  a  gas  from  the  fundamental  equa- 
tion. Returning  to  the  first  formula,  in  which  the  volume  of 
gas  is  unity,  the  product  nm  of  the  mass  of  one  molecule  by 
the  number  of  molecules  in  unit  volume  is  the  density  of  the 
gas.  The  pressure  p  corresponding  to  this  density  can  be 
measured,  and  hence  the  value  of  the  average  velocity  calcu- 
lated. Let  us  apply  this  equation  to  hydrogen  under  atmos- 
pheric pressure.  The  pressure  of  one  atmosphere  is  1013373 
dynes  per  square  centimetre  and  the  density  of  hydrogen  under 
this  pressure  and  at  the  temperature  of  melting  ice  is 
0.00008954  grammes  per  cubic  centimetre.  With  these  num- 
bers we  find  that  the  average  velocity  of  the  hydrogen  molecule 
is  184260  centimetres  per  second.  This  is  a  little  more  than  a 
mile  per  second.  From  Maxwell's  law  of  the  equality  of  the 
kinetic  energies,  the  velocity  of  the  molecules  of  any  other  gas 
is  to  the  velocity  of  the  hydrogen  molecule  inversely  as  the 
square  root  of  the  mass  of  the  molecule  of  the  gas  is  to  the 
square  root  of  the  mass  of  the  hydrogen  molecule.  Thus  if  the 
mass  of  the  oxygen  molecule  is  sixteen  times  that  of  the  hydro- 
gen molecule,  the  velocity  of  the  oxygen  molecule  is  one-quarter 
that  of  the  hydrogen  molecule. 


10-1 


FRICTION. 

79.  Resistance  to  Motion  Due  to  Friction. — When  a  block 
of  iron  or  wood  is  pushed  over  a  table,  the  moving  force  is 
always  opposed  by  a  resistance,  or  force  in  the  direction  oppo- 
site to  the  motion.     This  resisting  force  may  be  called  the 
force  of  friction,  or  simply  the  friction  between  the  two  bodies 
which  slide  over  each  other.    The  magnitude  of  the  friction  de- 
pends upon  the  nature  of  the  bodies  and  upon  the  state  of 
their  surfaces,  smooth  bodies  experiencing  less  friction  than 
rough  ones.     A  solid  of  any  form  moved  through  a  liquid,  or 
through  a  gas,  encounters  a  similar  resistance.     In  this  case 
the  parts  of  the  fluid  which  lie  nearest  to  the  solid  are  set  in 
motion  by  it,  and  as  they  in  turn  slide  past  other  parts  of  the 
fluid,   they   experience    friction   also,   and   communicate   their 
motion  to  those  parts,  so   that  the  work   which   is   done   in 
moving  the  solid  against  friction  is  largely  spent  in  moving 
the  parts  of  the  fluid.     When  the  motion  of  the  solid  ceases, 
the  moving  fluid  is  gradually  brought  to  rest  by  the  friction 
between  its  parts.    The  friction  between  the  parts  of  the  fluid 
is  called  internal  friction  or  viscosity.     The  fact  that  bodies 
moving  through  the  air  are  resisted  by  the  friction  of  the  air 
was  known  to  Galileo,  who  used  it  to  explain  the  fact  that 
very  light  bodies  with  large  sxirfaces  do  not  fall  at  the  same 
rate  as  heavy  compact  bodies  do.     This  resistance  of  the  air 
must  be  avoided,  or  taken  account  of,  in  exact  experiments  on 
the  laws  of  falling  bodies. 

80.  Friction    Between    Solids. — The    frictional    resistance 
which  a  solid  body  encounters  when  it  is  slid  over  a  solid  sur- 
face is-  believed  to  be  due  partly  to  the  direct  action  of  forces 
between  the  molecules  of  the  two  bodies,  and  partly  to  the 
partial  interlocking  of  the  small  protuberances  on  tlie  two 
surfaces.     The  latter  cause  is  more  prominent  when  the  sur- 
faces of  the  bodies  are  rough,  the  former  is,  at  least  rela- 


106  FRICTION. 

tively,  more  prominent   when  their   surfaces  are   smooth  or 
polished. 

The  laws  of  sliding  friction  were  investigated  by  Coulomb. 
One  of  the  two  bodies  which  he  employed  was  in  the  form  of 
a  long  horizontal  plane,  like  a  flat  board.  The  other  body 
was  pulled  over  it  by  a  weight,  and  the  observation  consisted 
in  determining  the  value  of  the  weight  required  to  give  the 
moving  body  a  constant  velocity  in  different  circumstances. 
He  found: 

1.  That  the  friction  between  two  bodies  is  proportional  to 
the  total  pressure  between  the  surfaces  in  contact. 

2.  That  the  friction  is  independent  of  the  extent  of  surface 
in  contact,  and  so  does  not  depend  directly  on  the  pressure 
per  unit  area. 

3.  That  the  friction  is  independent  of  the  velocity  of  the 
moving  bodies. 

Friction  is  also  exerted  between  two  bodies  which  roll  over 
each  other;  that  is,  friction  is  exerted  upon  a  wheel  rolling 
along  a  flat  surface.  Coulomb -found,  for  this  case  of  rolling 
friction,  that  the  friction  is  proportional  to  the  total  pressure, 
inversely  proportional  to  the  radius  of  the  wheel,  and  inde- 
pendent of  its  velocity. 

In  the  case  of  sliding  friction,  the  friction  or  the  resistance 
is  measured  by  the  weight  required  to  maintain  a  constant 
velocity  in  the  moving  body.  The  ratio  of  the  total  pressure 
on  that  body  to  this  resistance  is  called  the  coefficient  of  slid- 
ing friction.  The  coefficient  of  rolling  friction  is  given  by  an 
essentially  similar  definition. 

These  laws  of  friction  are  only  approximately  true.  It  has 
been  found  that  the  friction  falls  off  very  considerably  when 
the  velocity  of  the  moving  body  is  very  great.  Its  value  prob- 
ably changes  also  when  the  pressure  is  great.  The  coefficient 
of  friction  is  very  much  diminished  by  the  use  of  lubricants. 

81.  Friction  of  Liquids. — The  internal  friction  of  a  liquid 
was  discussed  theoretically  by  Newton,  on  the  assumption 
that  it  depends  on  the  relative  velocities  with  which  contigu- 
ous sheets  of  the  liquid  slide  past  each  other.  This  sort  of 


FRICTION.  107 

motion  is  that  which  is  produced  by  a  shearing  stress.  Since 
all  liquids  are  viscous3  that  is,  since  they  all  offer  more  or 
less  resistance  to  this  sort  of  motionj  it  cannot  be  said  that 
liquids  offer  no  resistance  to  a  shearing  stress.  The  resistance 
which  they  do  offer,  however,  depends  on  the  relative  motion 
of  their  parts,  and  since  liquids  do  not  possess  true  rigidity, 
any  shearing  stress  acting  in  them  will  produce  motion,  and 
the  liquid  will  gradually  assume  a  form  in  which  it  is  free 
from  shearing  stress. 

By  the  experimental  and  theoretical  study  of  the  effect  of 
friction  on  the  motion  of  a  solid  in  a  liquid,  or  of  the  motion 
of  a  liquid  flowing  pass  a  solid  wall,  or  through  a  tube,  it  has 
been  found  that  the  motion  is  affected  as  if  the  portions  of 
the  liquid  nearest  the  solid  adhered  to  it  without  slipping,  so 
that  the  resistance  to  the  motion  is  due  entirely  to  the  vis- 
cosity of  the  liquid.  When  the  floAV  of  a  liquid  through  long 
capillary  tubes  is  treated  on  this  assumption,  it  may  be  shown 
that  the  amount  of  liquid  which  issues  from  the  tube  in  unit 
of  time  is  proportional  to  the  fourth  power  of  the  radius  of 
the  tube.  This  theoretical  conclusion  is  in  complete  accord 
with  the  results  of  experiments  carried  out  by  Poisseuille. 

The  resistance  to  the  motion  of  a  solid  through  a  liquid 
increases  with  its  velocity.  When  a  force  of  a  given  value  is 
applied  to  a  body  to  move  it  through  a  liquid,  the  velocity  of 
the  body  will  gradually  increase  until  it  reaches  a  certain 
limiting  value,  for  which  the  resistance  due  to  friction  is  equal 
to  the  applied  force.  After  that  value  has  been  reached,  the 
velocity  will  remain  constant  so  long  as  the  force  is  applied. 
Thus  a  body  falling  through  water  will  have  a  constant 
velocity  after  it  has  fallen  a  certain  distance. 

82.  Friction  of  Gases. — In  general  what  has  been  said  about 
the  internal  friction  of  liquids  applies  also  to  gases.  The  pe- 
culiarity of  the  case  of  gases  is  this,  that  we  can  explain  their 
viscosity  by  means  of  the  kinetic  theory  of  gases.  When  one 
layer  or  sheet  of  a  gas  slides  past  another,  the  molecules 
which  dart  out  from  the  more  rapidly  moving  sheet  into  the 
other  one,  carry  with  them  a  certain  momentum  in  the  direc- 


108  FRICTION. 

tion  in  which  that  sheet  is  moving,  and  thus  communicate 
momentum  to  the  more  slowly  moving  sheet.  The  molecules 
which  dart  out  from  the  more  slowly  moving  sheet  into  the 
other  one  communicate  to  it  a  certain  momentum  in  the 
direction  opposite  to  its  motion.  Thus  the  momentum  of  the 
two  sheets  tends  to  become  the  same,  or  the  sheets  exert  a 
force  on  each  other.  Starting  with  this  conception,  the  laws 
of  internal  friction  iu  gases  can  be  deduced,  and  shown  to  be 
consistent  with  those  obtained  by  experiment.  In  particular, 
Maxwell,  to  whom  the  development  of  this  theory  is  due, 
showed  that  the  internal  friction  of  gases  should  be  inde- 
pendent of  their  densities.  The  experiments  which  Maxwell 
and  others  carried  out  to  test  this  conclusion  showed  that  it 
holds  true  within  very  wide  limits.  It  fails  only  when  the 
density  becomes  exceedingly  small. 

The  resistance  offered  by  a  gas  to  the  motion  of  a  body 
through  it  depends  on  the  velocity  of  the  body.  When  the 
velocity  is  small,  like  that  of  a  swinging  pendulum  or  a  mag- 
net, the  results  of  experiment  can  be  represented  by  supposing 
the  friction  to  be  proportional  to  the  velocity.  For  more 
rapidly  moving  bodies,  the  friction  seems  to  be  nearly  pro- 
portional to  the  square  of  the  velocity.  For  bodies  moving 
with  the  high  velocities  of  the  modern  rifle  ball  or  cannon  ball, 
the  friction  seems  to  be  even  greater  than  this  law  would  in- 
dicate. 

The  theory  of  the  motion  of  a  sphere  through  a  gas  leads 
to  the  conclusion  that  the  force  required  to  overcome  the  re- 
tarding effect  of  friction,  and  to  keep  the  sphere  moving  with 
a  constant  velocity,  is  proportional  to  the  radius  of  the 
sphere.  If  the  sphere  is  falling  through  the  gas,  the  force 
which  moves  it,  or  its  weight,  is  proportional  to  the  cube  of 
the  radius.  It  will  fall  with  continually  increasing  velocity 
until  it  reaches  at  last  a  limiting  velocity,  for  which  its  weight 
is  equal  to  the  resistance  offered  by  friction.  This  limiting 
velocity  is  therefore  proportional  to  the  square  of  its  radius. 
Thus  the  limiting  or  constant  velocity  attained  by  large  spheres 
falling  through  the  air  will  be  greater  than  that  attained  by 


FRICTION.  •        109 

small  spheres.  Large  drops  of  water  which  fall  from  a  cloud 
will  have  considerable  velocity  when  they  reach  the  ground, 
while  small  drops  will  move  toward  the  ground  more  slowly. 
The  very  small  drops  which  form  a  fog  or  cloud  will  fall  so 
slowly  that  their  motion  is  hardly  perceptible,  and  it  can 
easily  be  reversed  or  altered  by  the  motion  of  currents  of  air. 


no 


DIFFUSION. 

83.  Solution. — If  a  lump  of  sugar  is  dropped  into  a  vessel 
full  of  water,  it  gradually  disappears.  The  process  by  which 
it  disappears  seems  to  be  a  gradual  disintegration  of  those 
parts  of  it  which  are  nearest  the  water,  and  an  absorption  of 
them  into  it.  This  process  is  called  solution,  and  the  sugar 
is  said  to  dissolve  in  the  water.  After  solution  the  water  has 
acquired  certain  properties,  which  it  did  not  possess  before. 
These  are  supposed  to  be  due  to  the  presence  of  molecules  of 
sugar  distributed  throughout  the  water.  Very  many  bodies 
will  dissolve  in  water  or  in  other  liquids.  The  liquid  in  which 
solution  takes  place  is  called  the  solvent,  the  body  which  dis- 
solves in  it,  the  solute.  A  limited  amount  of  solvent  will  not 
dissolve  an  unlimited  amount  of  solute.  After  a  certain 
amount  has  been  dissolved,  the  limit  of  solution  has  been 
reached,  and  if  more  of  the  solute  is  present,  it  will  remain 
undissolved.  The  amount  of  solute  which  can  be  dissolved  in 
unit  mass  of  the  solvent  is  called  its  solubility. 

In  some  cases  there  seems  to  be  no  limit  to  the  amount  of 
solute  which  will  be  taken  up  by  a  solvent.  Thus  a  homo- 
geneous mixture  may  be  formed  by  adding  alcohol  or  glycerine 
in  any  quantity  to  a  limited  quantity  of  water.  In  cases  of 
this  sort,  we  speak  of  the  one  body  as  dissolved  in  the  other 
when  only  a  relatively  small  amount  of  it  is  used,  but  in  gen- 
eral the  two  bodies  are  said  to  be  mixed,  and  the  body  formed 
is  called  a  mixture. 

We  may  explain  the  act  of  solution,  in  a  general  way,  by 
supposing  that  the  molecular  forces,  acting  between  a  mole- 
cule of  the  solute  and  the  neighboring  molecules  of  the  solvent, 
are  strong  enough  •  to  tear  away  the  molecule  of  the  solute 
from  its  original  position  and  to  transfer  it  tp  the  solvent. 
It  is  uncertain  whether  the  molecule  of  the  solute,  after  it  has 
entered  the  solvent,  is  free,  or  whether  it  is  permanently  bound 
to  one  or  more  molecules  of  the  solvent,  so  as  to  form  what 


DIFFUSION  111 

is  called  a  molecular  aggregate.  The  weight  of  evidence  is 
at  present  in  favor  of  the  existence  of  such  molecular  aggre- 
gates in  many,  if  not  in  all,  solutions. 

84.  Free  Diffusion  of  Liquids. — If  a  quantity  of  sulphuric 
acid  is  placed  in  the  bottom  of  a  tall  cylindrical  jar,  and  if 
water  is  then  poured  carefully  into  the  jar,  in  such  a  way  that 
the  sulphuric  acid  below  is  not  disturbed,  the  water,  being 
relatively  lighter,  will  float  upon  the  acid,  and  the  surface  of 
separation  between  the  two  liquids  will  be  distinct.  In  pro- 
cess of  time,  this  distinct  surface  disappears,  and  a  region 
exists  between  the  pure  acid  below  and  the  pure  water  above 
in  which  there  is  a  mixture  of  water  and  acid.  This  region 
gradually  extends  until  at  last  it  occupies  the  whole  space 
filled  by  the  liquids.  At  first,  the  proportion  of  the  two 
liquids  found  in  different  parts  of  the  column  is  very  different, 
but  as  time  goes  on,  the  mixture  becomes  more  and  more 
homogeneous.  In  theory,  however,  it  cannot  become  entirely 
homogeneous  until  after  an  infinite  time  lias  elapsed.  This 
process,  by  which  one  liquid  mixes  w.itli  another,  while  the 
mixture  is  not  aided  by  currents  set  up  in  the  mass,  is  called 
diffusion,  and  a  case  of  this  particular  sort,  in  which  the  two 
liquids  are  not  separated  by  any  third  body  through  which 
they  must  pass  to  mix,  illustrates  free  diffusion. 

An  experiment  which  is  essentially  similar  may  be  tried 
by  placing  pure  water  upon  a  solution  of  copper  sulphate.  The 
dissolved  salt  will  gradually  rise  through  the  pure  water,  until 
the  whole  mass  becomes  a  solution  of  copper  sulphate  of 
uniform  strength. 

Any  two  liquids  which  may  be  mixed  will  diffuse  into  each 
other  in  the  way  here  described.  When  one  of  the  two  is  a 
solution,  we  consider  the  process  to  be  the  diffusion  of  the 
solute  from  the  parts  of  the  mass  in  which  its  concentration  is 
greater  to  the  parts  in  which  its  concentration  is  less.  It  was 
assumed  by  Fick,  and  his  assumption  has  been  verified  a? 
approximately  correct  in  many  cases,  that  the  rate  at  which 
diffusion  takes  place  along  a  line  is  proportional  to  the  rate 
of  change  of  concentration  along  that  line. 


112  DIFFUSION. 

AVe  may  explain  diffusion  by  ascribing  it  to  a  force  exerted 
upon  the  molecule  of  the  solute  by  the  molecules  of  the  sur- 
rounding solvent,  it  being  supposed  that  those  portions  of  the 
solvent  in  which  there  are  fewer  molecules  of  the  solute  exert 
the  greater  force.  The  motion  which  this  force  will  set  up  is 
resisted  by  the  friction  experienced  by  the  molecule  of  the 
solute  as  it  passes  through  the  solvent.  Certain  experiments 
which  have  been  made  indicate  that  this  friction  is  very  great. 

The  process  of  diffusion  may  also  be  explained  by  supposing 
it  to  be  due,  at  least  in  part,  to  the  motions  of  the  molecules 
of  the  solute.  We  have  every  reason  to  believe  that  these 
molecules  are  constantly  in  motion,  and  that  their  motion  is 
in  general  similar  to  that  already  considered  in  our  study  of 
gases.  It  is  easy  to  see  that  the  effect  of  such  motions  would 
be  to  distribute  the  molecules  of  the  solute  throughout  the 
whole  volume  to  which  it  has  access. 

85.  Diffusion  through  Membranes.  —  The  earliest  sys- 
tematic study  of  diffusion  was  made  on  the  diffusion  of  solu- 
tions through  solid  substances.  Many  substances,  as  for  ex- 
ample, unglazed  earthenware,  and  organic  bodies,  like  parch- 
ment paper  or  a  bladder,  or  even  india  rubber  in  thin  sheets, 
will  permit  liquids  to  diffuse  through  them.  If,  for  example, 
a  solution  of  common  salt  and  water  is  placed  in  a  bag  made 
of  parchment,  and  immersed  in  a  vessel  of  water,  the  salt  will 
gradually  pass  out  through  the  bag,  and  water  will  enter  it. 
If  another  salt  is  dissolved  in  the  water  outside  the  bag,  it 
will  generally  enter  the  bag  during  the  process.  The  rates 
at  which  the  two  salts  pass  through  the  membrane  depend  upon 
the  nature  of  the  salts  and  of  the  membrane.  The  general 
process  here  described  was  called  by  Dutrochet  osmosis. 

In  studying  the  phenomena  of  osmosis,  Graham  found  that 
bodies  differed  very  remarkably  with  respect  to  the  facility 
with  which  they  can  pass  through  an  organic  membrane.  He 
divided  all  soluble  bodies  into  two  classes,  to  which  he  gave 
the  names  crystalloid  and  colloid.  Crystalloid  bodies,  like 
sugar  or  salt,  are  those  which  in  their  solid  state  have  gen- 
erally a  recognizable  crystalline  form.  When  dissolved  in 


DIFFUSION.  113 

water,  they  diffuse  through  aii  organic  membrane  almost  as 
freely  as  if  the  membrane  were  not  present.  Colloid  bodies, 
like  gum-arabic  or  glue,  are  amorphous,  or  have  no  crystalline 
form.  They  scarcely  diffuse  at  all  through  an  organic  mem- 
brane. This  distinction  between  crystalloids  and  colloids  also 
appears  in  the  case  of  free  diffusion,  the  colloids  diffusing  very 
much  slower  than  the  crystalloids.  Graham  based  on  this 
difference  a  process,  called  dialysis,  of  separating  salts  from 
the  colloid  organic  bodies  with  which  they  may  be  mixed. 

86.  Laws  of  Osmotic  Pressure. — In  general,  as  has  already 
been  described,  osmosis  takes  place  in  both  directions  through 
an  organic  membrane,  and  with  such  membranes  no  very 
exact  or  general  laws  of  osmosis  can  be  discovered.  Traube 
discovered  that  membranes  can  be  obtained,  by  taking  advan- 
tage of  chemical  action,  which  in  many  instances  permit 
osmosis  in  only  one  direction.  Such  membranes  are  called 
semi-permeable  membranes.  If  a  small  mass  of  sulphate  of 
copper  is  dropped  into  a  dilute  solution  of  ferrocyanide  of 
potassium,  a  film  of  cyanide  of  copper  forms  over  its  exterior. 
Water  can  pass  by  osmosis 'through  this  film,  but  most  sub- 
stances which  dissolve  in  water  do  not  pass  through  it.  A 
film  of  this  sort  may  be  deposited  inside  the  pores  of  an  un- 
glazed  earthenware  jar.  If  the  jar  is  filled  with  a  solution  of 
sugar,  to  take  a  specific  example,  and  is  partially  immersed 
in  wrater,  the  sugar  will  remain  in  the  jar  and  water  will  enter 
the  jar  from  without.  The  membrane  is  permeable  to  water, 
but  is  not  permeable  to  sugar. 

To  study  osmosis  through  semi-permeable  membranes,  a 
small  jar  of  the  sort  just  described  is  closed  tightly  at  the 
mouth  by  a  stopper,  through  which  passes  a  long  glass  tube. 
This  arrangement  may  be  called  an  osmotic  cell.  It  was  used 
by  Pfeffer  in  his  researches.  If  such  a  cell  is  filled  to  the  stop- 
per with  a  solution  of  sugar,  or  of  any  ordinary  salt,  and  is 
immersed  in  water,  the  water  will  gradually  enter  the  cell,  so 
that  a  column  of  liquid  rises  in  the  tube.  This  process  may 
continue  for  many  days,  but  it  ceases  at  last  when  the  col- 
umn has  attained  a  height  depending  upon  the  concentration 


114  DIFFUSION. 

of  the  solution.  This  height  measures  the  pressure  which  will 
force  water  out  through  the  membrane  as  fast  as  it  is  coming 
in  by  osmosis.  The  pressure  thus  measured  is  called  the 
osmotic  pressure  of  the  solution.  If  we  arrange  an  osmotic 
cell  in  such  a  way  that  pressure  can  be  applied  to  the  surface 
of  the  solution  in  it,  and  so  adjust  this  pressure,  when  the  cell 
is  immersed  in  the  water,  that  water  neither  enters  nor  leaves 
the  cell,  the  pressure  thus  determined  is  the  osmotic  pressure. 

By  the  study  of  the  results  obtained  by  Pfeffer,  van't  Hoff 
has  shown  that  the  magnitude  of  the  osmotic  pressure  de- 
pends upon  the  amount  of  solute  in  tne  solution,  in  the  same 
way  exactly  as  the  pressure  of  a  gas  depends  upon  the  amount 
of  gas  enclosed  in  a  given  volume.  In  the  first  place,  the 
osmotic  pressure  of  a  solution  of  a  given  solute  is  proportional 
to  the  concentration  of  the  solution ;  that  is,  to  the  amount  of 
solute  in  unit  volume  of  the  solution.  If  we  consider  the  same 
quantity  of  solute  in  solutions  of  different  strengths,  it  is 
plain  that  the  coii«-i-ntr<tti<>iis  of  those  solutions  an-  invorsely 
as  their  volumes.  Thus  the  law  just  stated  is  equivalent  to 
the  statement  that  when  the  quantity  of  solute  is  fixed,  the 
os-motic  pressures  of  solutions  of  different  strengths  are  in- 
versely as  their  volumes.  That  is,  the  osmotic  pressure  obeys 
a  law  like  Boyle's  law. 

In  the  second  place,  for  a  solution  of  given  strength,  the 
osmotic  pressure  rises  with  the  temperature  according  to  the 
same  law  as  that  which  determines  the  relation  between  the 
pressure  and  temperature  of  a  gas.  That  is,  the  osmotic  pres- 
sure obeys  a  law  like  Gay-Lussae's  law. 

In  the  third  place,  the  osmotic  pressures  of  many  solutions 
are  equal  when  their  molecular  concentrations,  measured  by 
the  number  of  molecules  of  the  solute  contained  in  unit  vol- 
ume of  the  solution,  are  equal.  That  is,  for  these  solutions, 
the  osmotic  pressure  obeys  a  law  like  Avogadro's  law.  In 
many  other  cases,  "however,  this  law  ii?  not  fulfilled.  Arrhenius 
explained  this  fact  by  supposing  that  some  of  the  molecules 
of  the  solute,  in  these  cases,  are  disintegrated  into  their  con- 
stituent parts.  The  number  of  independent  portions  of  the 


DIFFUSION.  115 

solute — they  cannot  now  be  called  molecules — is  thereby  in- 
creased, and  the  osmotic  pressure  increases  likewise.  The 
molecules  thus  broken  up  are  said  to  be  dissociated  or  ionized. 
This  hypothesis  not  only  accounts  for  the  osmotic  pressures 
exhibited  by  such  solutions,  but  also  is  consistent  with  and 
explains  many  other  of  their  peculiarities. 

87.  Diffusion  of  Gases. — The  free  diffusion  of  gases  goes  on 
in  a  way  essentially  similar  to  that  of  liquids.  Its  funda- 
mental laws  have  been  developed  by  Maxwell  from  the  kinetic 
theory. 

The  diffusion  of  gases  through  porous  plates  of  earthen- 
ware, gypsum,  or  graphite,  was  studied  by  Graham,  who 
showed  that  the  rate  of  diffusion  is  proportional  to  the  differ- 
ence of  the  pressures  on  the  two  sides  of  the  porous  plate,  and 
is  inversely  proportional  to  the  square  roots  of  the  relative 
densities  of  the  gases.  Thus  oxygen,  which  is  sixteen  times 
as  heavy  as  hydrogen,  will  diffuse  only  one-quarter  as  fast. 
By  taking  advantage  of  this  fact,  we  may  partially  separate 
mixtures  of  different  gases.  These  laws  have  been  shown  by 
Reynolds  to  be  consistent  with  the  conclusions  of  the  kinetic 
theory. 

Gases  will  diffuse  through  thin  films  of  liquid,  like  the 
walls  of  a  soap  bubble.  They  will  dittuse  also  through  certain 
metals,  when  they  are  heated.  Thus  coal  gas  will  diffuse 
through  the  fire  pot  of  a  furnace,  if  it  becomes  red  hot. 


116 


SOUND. 

88.  General  Considerations  Respecting  Sound. — In  all  that 
we  have  studied  up  to  this  point  it  has  been  evident  that  the 
phenomena  considered  were  directly  presented  by  material 
bodies.  The  actions  of  these  bodies  on  one  another  are  forces 
which  cause  motions  or  bring  about  states  of  equilibrium.  We 
now  are  to  consider  another  set  of  phenomena,  associated  with 
the  sense  of  hearing,  which  at  first  sight  are  not  connected 
with  bodies  at  all.  These  phenomena  are  called  sounds. 
Although,  when  a  sound  is  heard,  there  is  nothing  in  the 
sensation  which  we  perceive  which  forces  us  to  associate  it 
with  the  action  of  any  external  material  bodies,  yet  universal 
experience  has  taught  us  that  sounds  are  associated  with  mat- 
ter. When  a  sound  is  heard,  we  always  believe  it  possible  to 
find  some  body  which  produces  the  sound,  that  is,  which  causes 
the  action  to  which  our  sensation  is  due;  and  we  further  be- 
lieve that  this  action  is  transmitted  to  our  ears  from  the  body 
which  originates  it  by  the  action  of  intermediate  bodies.  From 
the  time  of  Aristotle  it  has  been  commonly  believed  that  sound 
is  transmitted  through  the  air  as  a  series  of  disturbances  or 
impulses  communicated  to  it  by  the  body  which  is  the  source 
of  the  sound.  A  rival  theory,  which  assumed  that  sound  is 
transmitted  by  minute  particles  emitted  from  the  sounding 
body,  had  so  little  to  support  it  that  it  was  never  given  serious 
consideration  by  physicists.  All  subsequent  study  has  shown 
that  Aristotle's  theory  of  the  origin  and  the  mode  of  trans- 
mission of  sound  is  correct,  and  the  progress  of  knowledge  has 
resulted  simply  in  more  exact  statements  regarding  the  mode 
in  which  the  sounding  bodies  and  the  transmitting  medium 
act. 

Confusion  sometimes  arises  because  the  word  sound  is  used 
to  designate  a  sensation  received  through  our  sense  of  hearing, 
as  well  as  the  physical  action  by  which  that  sensation  is  ex- 
cited. In  what  follows  we  shall  commonly  use  the  word  to 


SOUND.  117 

denote  the  physical  action,  which  can  often  be  detected  even 
when  it  produces  no  audible  sound. 

Certain  sounds  produce  in  us  a  peculiar  sensation  which  so 
distinguishes  them  from  others  that  they  are  called  musical 
sounds  or  tones.  Such  musical  sounds  excite  a  sensation 
which  lasts  a  perceptible  time  without  perceptible  change.  In 
many  cases  also  the  sensation  produced  possesses  a  certain 
simplicity  which  distinguishes  it  from  sounds  of  other  sorts. 
Other  sounds,  which  do  not  possess  these  characteristics,  are 
called  noises.  We  shall  confine  our  attention  almost  altogether 
to  musical  tones. 

89.  Sounds  Produced  ~by  Strings. — The  use  of  stringed 
musical  instruments  is  of  great  antiquity.  It  must  have  been 
observed,  almost  as  soon  as  such  strings  were  used,  that  the 
tone  produced  by  the  string  depends  in  some  way  on  the  force 
with  which  the  string  is  stretched  and  on  the  length  of  the 
string.  It  is  credibly  reported  that  Pythagoras  investigated 
the  relation  between  the  length  of  a  string  and  the  pitch  of 
the  tone  emitted  by  it,  and  found  that  when  strings  whose 
lengths  were  to  each  other  as  one,  one-half,  two-thirds,  and 
three-fourths,  were  under  the  same  tension,  the  tones  emitted 
by  them  were  those  of  the  ordinary  musical  scale  called  the 
fundamental,  the  octave,  the  fifth,  and  the  fourth.  This  dis- 
covery of  Pythagoras  is  the  first  one  recorded  in  the  history 
of  physics,  antedating  by  more  than  two  hundred  years  any 
recorded  observations  in  mechanics. 

Galileo  and  his  friend  Mersenne  extended  this  observation 
of  Pythagoras  by  determining  the  lengths  of  string  which  pro- 
duced all  the  tones  of  the  ordinary  musical  scale.  They  also 
recognized  that  the  time  taken  by  the  string  to  execute  one 
vibration,  called  its  period,  or,  what  amounts  to  the  same 
thing,  the  number  of  vibrations  executed  by  the  string  in  one 
second,  which  is  the  reciprocal  of  the  period,  depends  on  the 
length  of  the  string.  They  supposed  that  the  period  is  directly 
proportional  to  the  length  of  the  string,  or  that  the  number 
of  vibrations  is  inversely  proportional  to  the  length  of  the 
string.  We  may  therefore  state  the  relation  which  they  de- 


118  SOUND. 

teimined  between  the  tones  of  the  musical  scale  and  the 
lengths  of  the  strings  which  produce  them  as  a  relation  be- 
tween those  tones  and  the  number  of  vibrations  producing 
them.  These  relations  are  given  for  the  major  and  minor 
scales  in  the  following  tables: 

MAJOR    SCALE. 

Name.  do       re         rni         fa        sol         la          si         do 

Number  of  n       -H         5n        4n        Sn        5™        15ra       2* 

vibrations.  8 "         4  3  2  3  8 

MINOR   SCALK. 

Name.  la       si         do        re         mi         fa        sol         la 

Number  of  n       _9»       _^_       4n        3n        Sn_      ^n_      ^ 

vibrations.  853256 

On  the  assumption  that  the  length  of  a  string  is  inversely 
proportional  to  its  number  of  vibrations^  Mersenne  deter- 
mined the  number  of  vibrations  of  a  string  emitting  a  partic- 
ular tone,  by  comparing  its  length  with  that  of  a  string  so 
long  that  its  vibrations  could  be  counted. 

In  all  that  has  been  said  so  far,  the  supposition  has  been 
made  that  the  strings  are  under  the  same  tension.  The  pitch 
of  the  tone  emitted  by  a  string  depends,  however,  not  only 
on  its  length,  but  on  its  tension,  on  the  material  of  which  it 
is  composed,  and  on  its  thickness.  Mersenne  found,  for  strings 
of  the  same  material,  that  the  number  of  vibrations  is  propor- 
tional to  the  square  root  of  the  tension,  is  inversely  propor- 
tional to  the  length,  and  is  inversely  proportional  to  the 
thickness.  Later  study  has  shown  that  if  strings  of  different 
materials  are  used,  the  number  of  vibrations  which  they  pro- 
d^uce  is  proportional  to  the  square  root  of  the  tension,  is 
inversely  proportional  to  the  lengtn,  and  is  inversely  propor- 
tional to  the  square  root  of  the  mass  of  the  string  in  unit 
length. 

90.  Transmission  of  Sound. — After  it  had  thus  been  shown 
that  a  musical  tone  originates  at  a  body  which  is  executing 
regular  vibrations,  attention  was  turned  to  the  question  of 


SOUND.  119 

the  transmission  of  these  vibrations.  They  are  manifestly 
transmitted,  in  most  cases,  through  the  air.  Other  bodies, 
however,  may  serve  as  the  medium  of  transmission.  Thus  the 
vibrations  of  a  piano  string  may  be  transmitted  to  the  audi- 
tory nerve  through  a  wooden  rod,  one  end  of  which  rests  on 
the  lid  of  the  piano,  while  the  other  end  is  held  in  the  teeth. 
The  vibrations  pass  through  the  rod  and  through  the  bones  of 
the  head,  without  passing  through  air.  A  scratch  of  a  pin, 
or  a  light  blow  of  a  hammer  on  one  end  of  a  long  log,  is 
heard  twice  by  a  person  at  the  other  end.  One  of  the  sounds 
is  transmitted  through  the  wood,  the  other  is  transmitted  less 
rapidly  through  the  air. 

The  problem  of  the  mode  of  transmission  of  sound  was 
first  successfully  studied  by  Newton.  He  assumed  that  the 
sounding  body  emitting  a  musical  tone  is  vibrating  in  such  a 
way  that  the  motion  of  each  of  its  parts  has  the  essential 
characteristics  of  the  motion  of  a  pendulum  bob.  That  is,  the 
acceleration  of  each  point  on  the  vibrating  body  is  propor- 
tional to  its  displacement  from  the  point  in  which  it  is  at  rest 
in  the  undisturbed  condition  of  the  body.  He  assumed  fur- 
ther, that  vibrations  of  a  similar  sort  are  transmitted  to  the 
particles  of  air  which  are  nearest  to  the  body,  from  these  to 
the  next  set  of  particles,  and  so  on.  He  showed  that,  on  these 
assumptions,  the  motion  of  any  one  particle  of  air  will  be  a 
vibratory  one  of  the  game  period  as  that  of  the  sounding  body, 
and  that  the  motion  transmitted  from  particle  to  particle  will 
proceed  outward  with  a  definite  velocity. 

We  may  describe  the  motion  in  the  air  by  which  sound  is 
transmitted,  in  the  following  way:  Consider  a  long  cylinder 
or  tube  filled  with  air,  and  suppose  that  a  layer  of  air  at  one 
end  of  it  is  pushed  forward.  As  it  advances  toward  the  next 
layer  of  air,  it  exerts  a  pressure  upon  it,  which  sets  that  layer 
in  motion.  The  second  layer  in  turn  sets  the  third  layer  in 
motion,  the  third  layer  the  fourth,  and  so  on;  and  thus  a 
forward  motion  is  transmitted  down  the  tube.  If  the  suc- 
cessive layers  of  air  were  rigidly  bound  together,  the  forward 
motion  of  the  first  layer  would  be  transmitted  instantaneously 


120  SOUND. 

through  the  whole  column.  In  fact,  the  transmission  of  the 
disturbance  is  not  instantaneous,  because  the  air  is  not  incom- 
pressible. The  first  layer  in  the  column  moves  up  toward  the 
second,  the  second  up  toward  the  third,  and  so  on,  producing 
a  region  in  which  the  air  is  more  condensed  than  it  is  in  its 
condition  of  equilibrium.  If  the  forward  motion  of  the  first 
layer  is  checked  after  it  has  moved  through  a  small  distance, 
this  process  of  condensation  ceases,  and  in  turn  the  other 
layers  take  up  positions  in  which  they  are  as  far  removed 
from  each  other  as  they  were  before  the  disturbing  motion 
took  place.  The  condensation  originally  produced  proceeds 
as  a  form  of  motion  along  the  column. 

If  the  first  layer  is  riow  suddenly  drawn  backward  to  its 
first  position,  the  second  layer  is  pushed  toward  it  by  the  pres- 
sure on  the  other  side  of  it,  the  third  layer  is  pushed  toward 
the  second,  and  so  on.  The  movements  of  these  layers  like- 
wise take  place  successively,  so  that  while  they  are  going  on, 
the  air  between  the  first  layer  and  a  part  of  the  column  which 
the  disturbance  has  not  yet  reached  occupies  a  larger  volume 
than  it  does  in  its  undisturbed  condition,  or  is  rarefied.  When 
the  first  layer  has  come  to  rest,  the  other  layers  successively 
come  to  rest  in  the  positions  which  they  originally  occupied, 
and  the  rarefaction  which  has  been  produced  proceeds  up  the 
column. 

Now,  if  the  first  layer  of  air  is  forced  to  execute  regular 
vibrations,  like  those  of  a  pendulum  bob,  it  will  produce  con- 
densations and  rarefactions  similar  to  those  which  have  been 
described,  which  will  succeed  each  other  at  regular  intervals 
and  proceed  up  the  column  at  a  uniform  rate  as  sound  waves. 
The  time  in  which  the  exciting  vibration  is  executed,  or  the 
time  between  the  production  of  two  conditions  of  maximum 
condensation,  is  called  the  period  of  the  sound.  The  distance 
between  two  successive  points  of  maximum  condensation  is 
the  wave  length. 

When  a  sound  is  produced,  as  is  commonly  the  case,  in  a 
body  of  air  extending  in  all  directions,  the  successive  con- 
densations and  rarefactions  proceed  outward  from  the  body 


SOUND.  121 

in  all  directions,  and  a  particular  condensation  or  rarefaction, 
which  has  left  the  body  at  any  instant,  will  be  found,  at  a 
later  instant,  on  the  surface  of  a  sphere  whose  centre  is  in  the 
sounding  body.  The  analogy  between  these  spherical  sound- 
waves and  the  circular  waves  which  are  set  up  on  the  surface 
of  still  water,  by  a  slight  disturbance  made  at  a  point  in  it, 
was  perceived  by  Yitnwius,  although  of  course  he  had  no 
exact  knowledge  of  the  character  of  sound-waves. 

If  we  fix  on  the  most  condensed  portion  of  a  particular 
wave  as  a  characteristic  point  in  it,  we  may  define  the  velocity 
of  a  wave  as  the  velocity  of  its  most  condensed  portion.  It  ii 
equal  to  the  wave  length  divided  by  the  period.  This  velocity 
will  plainly  depend  on  the  rate  at  which  the  successive  layers 
of  air  yield  to  the  unequal  pressure  on  the  two  sides  of  them 
and  this  depends  on  the  elasticity  of  the  air  and  on  the  mase 
of  the  particles  which  are  to  be  moved,  or  on  the  density  ot 
the  air.  Newton  showed  that  the  velocity  of  a  sound-wave 
not  only  in  air,  but  in  any  medium,  is  equal  to  the  square  root 
of  the  elasticity  divided  by  the  square  root  of  the  density 
In  Newton's  time,  the  elasticity  of  a  medium  which,  like  air. 
obeys  Boyle's  law,  was  thought  to  be  equal  to  its  pressure 
When  Newton  used  this  value  of  the  elasticity  to  calculate  the 
velocity  of  sound,  he  obtained  a  value  for  it  which  was  less 
than  that  which  had  been  obtained  by  direct  experiment.  This 
discrepancy  between  theory  and  experiment  was  removed 
many  years  later  by  Laplace,  who  called  attention  to  the  fact 
that  the  elasticity  of  air  is  equal  to  its  pressure  only  when 
the  temperature  of  the  air  is  kept  constant.  Now  it  is  known 
that  a  sudden  condensation  of  air  will  raise  its  temperature 
and  a  sudden  rarefaction  will  lower  its  temperature.  The 
condensations  and  rarefactions  which  constitute  a  sound-wave 
pass  through  the  air  so  rapidly  that  no  time  is  given  for  the 
equalization  of  the  differences  of  temperature  which  they 
produce,  so  that  the  elasticity  which  is  to  be  used  in  the  cal 
culation  of  the  velocity  of  sound  is  one  determined  on  the 
condition  that  heat  is  neither  received  nor  emitted  by  the 
air.  The  value  of  this  elasticitv  for  air  was  measured  and 


122  SOU  ML> 

found  to  be  equal  to  its  pressure,  multiplied  by  the  numerical 
factor  1.41.  With  this  value  of  the  elasticity,  the  calculated 
velocity  of  sound  agreed  with  the  experimental  value. 

To  calculate  the  velocity  of  sound,  we  have  given  the 
numerical  factor  1.41,  the  pressure  of  one  atmosphere,  1013373 
dynes  per  square  centimetre,  and  the  density  of  air  under  the 
pressure  of  one  atmosphere  and  at  the  temperature  of  melting 
ice,  0001^3  grammes  per  cubic  centimetre.  From  these 
data,  we  calculate  that  the  velocity  of  sound  equals  332.4 
metres  per  second.  The  density  of  air  equals  its  mass  con- 
tained in  a  given  volume  divided  by  that  volume.  If  we  sub- 
stitute this  value  of  the  density  in  the  formula  for  the  veloc- 
ity, we  obtain  the  result  that  the  velocity  is  equal  to  the 
square  root  of  the  product  of  the  pressure  and  the  volume  of 
a  given  mass  of  air,  multiplied  by  a  factor  which  is  a  con- 
stant. Now  from  Boyle's  law  we  know  that  the  product  of 
the  pressure  and  the  volume  of  a  given  mass  of  air  is  con- 
stant. Therefore,  the  velocity  of  sound  in  air  will  be  the 
same,  at  the  same  temperature,  whatever  be  its  density.  Thus 
it  will  be  the  same  at  the  highest  altitudes  as  it  is  at  the  sea 
level.  From  the  same  formula  we  learn  that  the  velocity  of 
sound  in  different  gases  is  inversely  proportional  to  the  square 
roots  of  their  densities.  Thus  in  a  light  gas,  like  hydrogen 
or  coal  gas,  the  velocity  is  greater  than  it  is  in  air. 

The  velocity  of  sound  was  determined  experimentally  by 
Mersenne,  by  noting  the  time  which  elapsed  between  the  in- 
stant at  which  the  flash  of  a  pistol  was  seen  and  the  instant 
at  which  the  report  was  heard.  By  using  sounds  of  different 
intensities,  produced  by  a  cannon  and  a  pistol,  Gassendi  proved 
that  the  velocity  of  sound  is  independent  of  its  loudness.  That 
it  is  also  independent  of  its  pitch  is  shown  by  the  fact  that 
the  different  tones  produced  by  the  instruments  of  a  band  or 
orchestra  do  not  lose  their  harmonious  relations  when  heard 
by  a  distant  observer.  The  statement  that  the  velocity  of 
sound  is  independent  of  its  loudness  is  not  absolutely  correct. 
Captain  Parry  relates  that  when  his  men  were  engaged  in  gun 
practice  in  the  still  air  of  the  Arctic  regions,  the  report  of  a 


*  IUND.  123 

gun  was  heard  by  a  very  distant  observer  before  he  heard  the 
word  of  command  to  fire.  A  profound  study  by  Earnshaw  of 
the  propagation  of  waves  through  air  has  shown  that  very  loud 
sounds  should  be  propagated  with  greater  velocities. 

The  mean  of  three  of  the  best  experimental  determinations 
of  the  velocity  of  sound  at  0°  Cent,  gives  the  value  331.5 
metres  per  second. 

91.  Reflection  of  tiound-Waves. — When  a  sound-wave  in 
air  meets  a  solid  obstacle,  like  a  high  wall,  it  is  reflected  from 
it;  that  is,  it  is  turned  back  in  its  course  and  proceeds  from 
then  on  in  a  reverse  direction.  From  the  analogous  reflection 
which  may  easily  be  observed  when  the  waves  on  water  en- 
counter an  obstacle,  Vitruvius  supposed  such  a  reflection  to 
take  place  in  the  case  of  sound,  and  explained  thereby  the 
echo.  Observation  shows  that  the  sound-wave  does  not  lose 
its  essential  characteristics  by  reflection. 

Reflection  of  sound  occurs  whenever  the  wave  meets  a  sur- 
face which  separates  the  medium  in  which  it  is  travelling 
from  another  one.  Two  cases  of  reflection  must  be  distin- 
guished, which  depend  on  the  relations  of  the  motions  of  the 
two  media  when  they  are  transmitting  similar  sounds.  If 
the  second  medium  is  made  up  of  particles  which  move  over 
longer  distances  than  those  of  the  first  medium  do,  when  they 
are  transmitting  similar  sounds,  and  if  we  consider  that  por- 
tion of  the  advancing  wave  in  which  the  particles  of  tha 
medium  are  being  pushed  forward,  it  is  plain  that  when  it 
reaches  the  surface  of  separation,  the  last  layer  of  particles  of 
the  first  medium  will  move  farther  out  into  the  secpnd  than 
they  would  have  done  if  the  first  medium  had  continued  fur- 
ther. With  respect  to  the  first  medium,  this  movement  of  the 
last  layer  may  be  looked  on  as  composed  of  two  movements, 
one  equal  to  that  which  the  other  particles  of  the  first  medium 
have  executed,  and  an  additional  movement  in  the  same  direc- 
tion. This  additional  movement  is  of  such  a  sort  as  to 
start  a  movement  in  the  first  medium,  proceeding  in  the  op- 
posite direction  to  that  of  the  original  wave.  The  successive 
displacements  caused  by  this  movement  are  in  the  same  direc- 


124  SOUND. 

tion  in  space,  or  have  the  same  sign,  as  the  displacements  in 
the  original  movement  by  which  they  were  produced.  This 
sort  of  reflection  is  therefore  called  reflection  without  change 
of  sign. 

On  the  other  hand,  if  the  second  medium  is  one  whose 
particles  move  less  freely  than  those  of  the  first  medium,  a 
forward  displacement  is  diminished  when  it  comes  to  the 
surface  of  separation.  We  may  look  on  the  forward  displace- 
ment of  the  particles  of  the  first  medium  at  that  surface  as 
composed  of  two  displacements,  one  equal  to. the  original  for- 
ward displacement,  the  other  less  than  it  in  amount  and  in 
the  opposite  direction.  This  displacement  in  the  opposite 
direction  is  of  such  a  sort  as  to  produce  a  movement  in  the 
first  medium,  proceeding  in  a  direction  opposite  tq  that  of  the 
original  wave.  In  this  case  the  reflection  is  said  to  be  reflec- 
tion with  change  of  sign. 

These  two  kinds  of  reflection  are  important  in  connection 
with  the  study  of  the  vibrations  of  sounding  bodies. 

92.  Resonance. — It  was  observed  by  Mersenne  that  if  two 
strings  are  tuned  to  the  same  pitch,  and  one  of  them  is  sounded, 
the  other  will  take  up  a  vibration  and  will  also  emit  a  sound. 
The  experiment  succeeds  especially  well  if  the  two  strings  are 
mounted  on  the  same  sounding  board.  It  was  also  observed 
that  a  string  will  be  set  in  vibration  if  another  string  is 
vibrating  near  it  which  is  tuned  to  the  octave  or  to  the 
twelfth  of  the  tone  which  it  will  emit.  This  general  phenom- 
enon of  the  excitation  of  vibrations  in  bodies  by  the  vibrations 
of  other  bodies  in  their  presence  is  called  resonance. 

The  phenomenon  of  resonance  is  not  exhibited  by  string* 
only,  but  by  all  sorts  of  sounding  bodies.  Thus,  one  tuning 
fork  will  respond  to  the  vibrations  of  another  which  is  in 
unison  or  in  simple  harmony  with  it,  and  a  column  of  air  will 
respond  to  the  vibrations  of  a  tuning  fork  or  of  a  string. 

To  explain  resonance  we  consider  the  effect  of  a  succession 
of  small  impulses  applied  to  a  string,  which  is  capable  of 
regular  vibration,  at  intervals  which  are  equal  to  the  natural 
period  of  vibration.  Such  impulses  are  applied  to  it  by  the 


SOUND.  125 

motions  of  the  air,  or  of  some  vibrating  body,  like  the  sound- 
ing board  which  supports  it,  and  these  are  set  up  by  the  regu- 
lar vibrations  of  the  first  sounding  body.  The  effect  of  the 
first  impulse  is  a  slight  disturbance  of  the  string.  If  that 
impulse  were  the  only  one  applied  to  it,  it  would  execute  a 
number  of  minute  vibrations  in  its  own  natural  period.  A 
second  impulse,  however,  is  applied  at  the  end  of  that  period, 
and  therefore  just  at  the  time  when  the  string  is  moving 
through  the  position  from  which  it  was  started  by  the  first 
impulse.  The  effect  of  the  second  impulse  is  added  to  that  of 
the  first,  and  the  vibration  of  the  string  is  increased  thereby. 
As  the  successive  impulses  are  applied  to  it,  each  at  the  time 
when  its  action  will  increase  the  vibration,  the  vibrations  be- 
come greater  and  greater  until  they  can  often  be  perceived  by 
the  eye  and  can  be  heard  to  give  forth  a  musical  tone. 

If  the  natural  period  of  the  string  is  not  equal  to  the  in- 
terval between  the  impulses,  the  effect  of  the  first  impulse  will 
again  be  to  start  a  small  vibration,  but  each  impulse  following 
the  first  will  be  applied  at  a  time  which  departs  more  and 
more  from  that  at  which  the  string  is  in  such  a  position  that 
its  vibrations  are  increased  by  the  impulses.  For  example,  if 
the  string  naturally  executes  200  complete  vibrations  in  a 
second,  and  if  the  sounding  body  which  sends  impulses  to  it 
executes  201  complete  vibrations  in  a  second,  the  second  im- 
pulse is  applied  one  two-hundredth  of  a  second  before  the 
natural  vibration  of  the  string  brings  it  to  its  most  favorable 
position,  the  third  impulse  is  applied  two  two-hundredths  of 
a  second  too  soon,  and  so  on,  the  successive  impulses  being  ap- 
plied at  times  in  which  the  position  of  the  string  is  less  and 
less  favorable.  After  100  vibrations  have  been  executed,  the 
impulse  which  tends  to  move  the  string  in  one  sense  is  applied 
to  it  just  when  it  is  moving  in  an  opposite  sense,  and  so  its 
effect  is  to  destroy  the  original  vibration.  Thus  the  string 
will  not  respond,  and  resonance  will  not  occur,  unless  the 
string  has  the  same  period  of  vibration  as  the  sounding  body. 

When  a  string  is  made  to  respond  to  the  vibrations  of  a 
body  which  is  emitting  the  octave  of  the  fundamental  tone 


J26  SOUND. 

which  the  string  emits,  the  tone  emitted  by  the  string  is  also 
the  octave.  In  this  case,  as  was  shown  by  Noble  and  Pigot, 
and  afterwards  by  Sauveur,  the  string  is  vibrating  in  two 
portions.  .  Its  middle  point  is  at  rest,  and  each  half  of  it 
vibrates  as  if  it  were  an  independent  string,  with  its  natural 
period  of  vibration.  When  the  resonance  of  a  string  is  pro- 
duced by  a  body  sounding  the  twelfth  of  the  fundamental 
tone  of  the  string,  the  responding  string  vibrates  in  three 
equal  parts.  At  two  points,  which  divide  it  into  these  three 
equal  parts,  the  string  is  at  rest. 

These  points  at  which  the  string  is  at  rest,  when  it  is 
emitting  a  tone  which  is  higher  than  its  fundamental,  are 
called  nodes.  The  vibrating  portions  between  the  nodes  are 
called  ventral  segments  or  loops.  The  tones  which  a  string 
will  emit,  when  it  vibrates  in  parts  which  are  fractional  parts 
of  its  whole  length,  are  called  harmonics,  partial  tones,  or 
overtones. 

An  acute  ear  may  detect  that  when  a  string  is  emitting  its 
fundamental  tone,  it  is  also  emitting  some  of  its  overtones. 
This  observation  was  made  by  the  early  observers  whose 
names  have  already  been  mentioned.  It  shows  that  the  vibra- 
tion of  a  string  may  be  much  more  complicated  than  the  simple 
vibration  which  we  have  assumed  so  far  to  be  the  one  executed 
by  it. 

In  general,  what  has  been  said  of  a  string  with  respect  to 
resonance  and  with  respect  to  the  production  of  overtones,  not 
only  singly,  but  in  combination  with  fundamental  tones,  may 
be  said  also  of  all  sorts  of  sounding  bodies. 

93.  Sounding  Bodies. — The  phenomena  which  have  now 
been  described  directed  attention  to  the  mode  of  vibration  of 
sounding  bodies.  It  was  at  first  assumed,  as  we  have  so  far 
done,  that  the  vibration  of  each  part  of  the  body  is  a  motion 
similar  to  that  of  a  pendulum  bob,  or,  as  we  may  call  it,  a 
simple  harmonic  motion.  The  fundamental  characteristic  of 
such  a  motion,  wThich  takes  place  in  a  straight  line,  or  in  a 
curve  so  slight  that  it  may  be  considered  a  straight  line,  is 
that  the  acceleration  of  the  body  is  always  proportional  to 


SOUND.  127 

its  displacement  from  its  position  oi  equilibrium,  in  which  it 
lies  when  the  body  is  at  rest.  This  kind  of  motion  will  be  set 
up  in  the  body  by  its  elastic  forces,  when  the  deformation  of 
the  body  is  not  great;  for,  as  has  already  been  seen  in  the 
study  of  elasticity,  Hooke's  law  holds,  for  all  sorts  of  small 
displacements,  that  the  elastic  force  excited  is  proportional 
to  the  displacement. 

The  simultaneous  production  of  a  fundamental  tone  and 
of  one  or  more  overtones  by  the  same  body,  forces  us  to 
assume  that  the  vibrations  in  it  are  not  simple  harmonic 
motions,  it  was  shown  by  Daniel  Bernoulli  that  in  this  case 
the  vibration  of  a  part  of  the  body  may  be  considered  as  aris- 
ing from  the  coexistence  in  it  of  several  simple  harmonic 
motions,  whose  periods  correspond  to  those  of  the  different 
tones  which  the  body  is  producing.  The  result  of  this  com- 
bination of  simple  harmonic  motions  is  a  periodic  motion ; 
that  is,  a  motion  which,  however  complicated  it  may  be,  re- 
peats itself  over  and  over  again  in  successive  equal  periods. 
In  the  case  of  a  sounding  body,  this  period  is  that  of  the 
fundamental  tone. 

To  prove  this,  Bernoulli  demonstrated  the  principle  of  the 
superposition  of  small  motions.  This  principle  may  be  de- 
scribed as  follows:  If  a  part  of  an  elastic  body  is  set  in 
motion  by  two  or  more  disturbances  reaching  it  at  the  same 
instant,  its  displacement  will  be  the  resultant  of  the  displace- 
ments which  those  disturbances  would  impart  to  it  if  they 
were  to  affect  it  separately;  and  the  disturbances  which  it 
will  cause  in  the  parts  of  the  body  which  lie  near  it  are  also 
the  resultants  of  those  which  it  would  have  caused  if  it  had 
been  affected  by  each  of  these  disturbances  separately.  We 
may  illustrate  this  principle  in  certain  special  cases.  For 
example,  if  a  sound  wave  is  produced  on  one  side  of  a  room, 
it  will  pass  across  the  room  and  affect  a  portion  of  the  air, 
in  a  certain  place  and  at  a  given  instant,  in  a  certain  way. 
The  same  portion  of  air  will  be  affected  also  and  differently 
by  another  sound-wave  starting  from  the  other  side  of  the 
room.  If  both  waves  are  set  up  so  as  to  reach  the  chosen  por- 


tion  of  air  at  the  same  instant,  it  will  Le  disturbed  by  both  of 
them,  so  that  its  resultant  motion  is  a  combination  of  the  two 
motions  which  they  will  separately  produce.  The  two  compo- 
nents of  this  motion  affect  the  air  around  this  portion  inde- 
pendently, and  each  of  the  two  waves  travels  on  from  the 
point  at  which  they  cross  each  other  without  being  changed 
in  any  way.  As  another  example,  let  us  consider  a  point  at 
the  middle  of  a  vibrating  string,  at  which  two  disturbances 
from  the  opposite  ends  of  the  string  arrive  at  the  same 
instant.  These  two  disturbances  may  be  such  as  to  make  the 
resultant  disturbance  of  the  point  double  that  which  it  would 
acquire  from  either  one  of  them,  or  they  may  be  such,  being 
of  the  same  magnitude  and  in  opposite  directions,  that  the 
resultant  disturbance  at  the  point  is  zero,  or  the  resultant 
may  have  any  intermediate  value.  In  any  case,  after  the  dis- 
turbances have  passed  the  point,  they  will  proceed  unchanged 
by  any  action  of  the  one  upon  the  other. 

By  the  use  of  this  principle  of  the  superposition  of  small 
motions,  and  of  the  laws  of  reflection,  we  may  give  a  general 
explanation  of  the  vibrations  of  sounding  bodies.  The  general 
characteristic  of  such  vibrations  is  that  they  take  the  form  of 
what  are  called  standing  or  stationary  waves  in  the  body. 
We  may  illustrate  the  formation  of  such  waves  by  a  special 
example.  Suppose  that  a  stretched  string  is  plucked  near  one 
of  its  ends.  The  disturbance  produced,  which  we  may  call  an 
elevation,  runs  along  the  string  to  the  other  end  as  a  simple 
wave,  where  it  meets  the  rigid  support  in  which  that  end  of 
the  string  is  held.  It  is  therefore  reflected  with  change  of 
sign  and  comes  back  as  a  depression.  It  is  again  reflected  at 
the  end  from  which  it  started,  and  becomes  there  an  elevation, 
which  proceeds  along  the  string  again  as  has  already  been 
described.  The  string  is  thus  set  in  vibration.  If  the  part  of 
the  string  which  was  first  plucked  is  plucked  again  and  again, 
at  times  coinciding  with  those  of  the  return  of  the  reflected 
wave  to  the  place  of  origin,  the  successive  disturbances  will 
be  superposed  and  the  vibration  will  be  increased.  If  the 
impulses  are  applied  to  the  string  twice  as  often,  so  that  the 


SOUND.  129 

advancing  wave  reaches  the  other  end  of  the  string  just  as 
the  second  impulse  is  applied,  the  elevation  produced  by  the 
second  impulse,  and  the  depression  due  to  the  first  impulse 
after  reflection,  will  meet  in  the  middle  of  the  string,  and  the 
point  at  which  they  meet  will  not  be  disturbed.  It  will 
remain  undisturbed  as  the  successive  elevations  and  depres- 
sions pass  through  it,  while  the  parts  of  the  string  on  either 
side  of  it  will  vibrate.  If  the  impulses  are  applie'd  three  times 
as  often  as  in  the  first  case,  there  will  be  two  points,  dividing 
the  string  into  three  equal  parts,  at  which  the  advancing  ele- 
vations and  returning  depressions  will  meet,  and  which  will 
consequently  remain  at  rest.  Similar  statements  may  evi- 
dently be  made  for  disturbances  applied  more  frequently  still, 
if  only  the  time  between  them  is  some  fractional  part  of  the 
time  required  by  the  original  disturbance  to  traverse  the 
length  of  the  string.  The  points  at  which  the  string  is  at  rest 
are  the  nodes,  the  parts  between  them,  the  ventral  segments. 
The  disturbance  on  the  string  by  which  its  parts  are  kept  in 
uniform  vibratory  motion,  is  called  the  standing  or  stationary 
wave.  The  disturbance  usually  impressed  upon  a  string  is  not 
an  impulse  at  one  point,  but  a  general  displacement  of  a  con- 
siderable portion  of  the  string,  or  even  of  the  whole  string  at 
once.  By  the  vise  of  the  principle  of  superposition,  we  see 
that  the  disturbance  of  each  part  of  the  string  will  be  propa- 
gated independently  in  the  way  already  described  for  a  simple 
disturbance,  and  nodes  will  be  produced  when  the  period  of  the 
original  disturbance  has  any  one  of  the  values  already  indi- 
cated in  the  preceding  discussion. 

In  order  to  develop  standing  waves,  it  is  not  necessary  to 
apply  to  the  string  impulses  which  are  properly  timed.  It  is 
sufficient  to  apply  a  large  number  of  impulses,  among  which 
there  will  be  many  which  occur  at  the  proper  times.  As  has 
already  been  explained,  in  §  92,  the  string  will  respond  to  those 
impulses,  and  will  not  be  affected  by  the  others.  For  example, 
when  a  bow  is  drawn  across  a  violin  string,  the  impulses  which 
it  applies  to  the  string  have  no  regularity,  yet  the  string 
responds  to  those  which  occur  at  intervals  corresponding  to  its 


130  SOUND. 

fundamental  period,  and  also  to  those  occurring  at  intervals 
corresponding  to  the  periods  of  some  of  its  overtones.  Those 
overtones  cannot  occur  which  would  establish  nodes  or  fixed 
points  in  that  part  of  the  string  which  is  kept  moving  by  the 
bow. 

The  fundamental  tone  of  a  sounding  body  is  determined  by 
the  longest  complete  part  of  a  standing  wave,  which  can  be 
set  up  in  it.  The  overtones  which  it  will  emit  depend  upon 
the  particular  shorter  standing  waves  which  can  be  set  up  in 
it.  We  may  illustrate  this  general  statement  by  the  consider- 
ation of  certain  special  sounding  bodies. 

The  vibrating  string,  which  we  have  already  described,  is 
fixed  at  both  ends,  so  that  reflection  occurs  at  both  ends  with 
change  of  sign.  The  time  required  by  the  string  to  execute  a 
complete  vibration,  or  the  period  of  the  fundamental  tone 
which  it  will  emit,  is  the  time  required  by  the  disturbance  set 
up  at  one  end  of  the  string  to  pass  from  its  point  of  origin  to 
the  other  end  of  the  string  and  back  again.  That  is,  the  period 
of  the  fundamental  tone  is  equal  to  twice  the  time  required 
for  a  wave  to  travel  over  the  length  of  the  string.  The  string 
is  half  the  length  of  the  fundamental  standing  wave  in  it.  Tt 
may  easily  be  seen  that  a  node  may  occur  in  the  middle  of  the 
string,  and  that  the  vibrations  of  the  halves  of  the  string  will 
occur  in  half  the  fundamental  period,  so  that  the  string 
vibrating  in  this  way  gives  the  octave  of  the  fundamental. 
The  string  may  also  be  divided  by  nodes  into  three  parts,  four 
parts,  or  any  number  of  fractional  parts.  To  each  of  these 
modes  of  subdivision  there  corresponds  a  tone  which  the  string 
can  produce.  Thus  a  stretched  string  will  produce  a  funda- 
mental tone  and  all  possible  overtones. 

A  column  of  air  in  a  pipe  may  have  standing  waves  set  up 
in  it  by  a  succession  of  properly  timed  impulses  applied  at 
one  end  of  the  column,  or  if  a  number  of  differently  timed  im- 
pulses are  applied,  as  is  clone  in  the  case  of  the  organ  pipe  by 
blowing  against  the  lip  of  the  pipe,  the  column  will  respond 
to  certain  of  those  impulses  and  standing  waves  will  be  set  up 
in  it.  If  the  pipe  is  open  at  both  ends,  the  wave  which  ad- 


SOUND.  131 

vances  from  the  bottom  of  the  pipe  to  the  top,  is  reflected  at 
the  top  without  change  of  sign.  Thus  an  upward  displacement 
which  leaves  the  bottom  of  the  pipe  is  reflected  as  an  upward 
displacement.  A  standing  wave  will  be  established  in  the 
column  if  the  displacement  which  returns  from  the  first  re- 
flection and  is  again  reflected  at  the  bottom  as  an  upward  dis- 
placement is  superposed  on  the  next  upward  displacement 
produced  by  the  body  which  is  setting  up  the  disturbance.  The 
time  between  the  two  displacements  is  therefore  the  time 
required  for  the  wave  to  travel  twice  the  length  of  the  pipe. 
The  wave  length  in  air  of  the  fundamental  tone  which  is 
produced  by  un  open  pipe  is  therefore  twice  the  length  of  the 
pipe.  The  returning  upward  displacement,  when  it  reaches 
the  middle  of  the  pipe,  meets  there  an  advancing  displacement 
in  the  opposite  sense,  or  a  downward  displacement,  which  has 
left  the  lower  end  of  the  pipe  at  a  time  later  by  half  a  period 
than  that  at  which  the  original  displacement  left  the  same 
place.  These  two  displacements  combine  to  produce  no  dis- 
placement at  the  middle  of  the  pipe,  and  the  special  result 
here  described  holds  generally  for  all  the  disturbances  which 
start  from  the  lower  end.  They  will  always  meet  in  the  middle 
of  the  pipe  with  reflected  disturbances  which  are  equal  and 
opposite  to  them.  There  will  therefore  be  a  node  at  the  middle 
of  the  pipe,  when  it  is  giving  its  fundamental  tone. 

If  the  impressed  vibrations  occur  twice  as  often  as  those 
which  would  give  the  fundamental  tone,  an  upward  displace- 
ment will  reach  the  upper  end  of  the  pipe  just  as  the  next 
upward  displacement  is  produced  at  the  bottom.  At  the  same 
instant,  a  downward  displacement  will  exist  in  the  middle  of 
the  pipe.  This  downward  displacement  moving  up  the  pipe  will 
meet  the  reflected  upward  displacement  at  a  point  one-quarter 
of  the  length  of  the  pipe  distant  from  the  top.  A  node  there- 
fore exists  at  that  point.  The  reflected  upward  displacement, 
proceeding  further  down  the  pipe,  will  meet  another  advancing 
downward  displacement  at  a  point  one-quarter  the  length  of 
the  pipe  from  the  bottom.  There  is  therefore  a  node  at  this 
point  also.  The  period  of  the  tone  emitted  by  this  vibration 


132 

is  equal  to  the  time  taken  by  a  wave  to  traverse  the  length 
of  the  pipe.  The  wave  length  is  equal  to  the  length  of  the 
pipe.  The  tone  due  to  this  vibration  is  the  octave  of  the 
fundamental. 

Other  vibrations  will  set  up  standing  waves  in  such  a  pipe, 
whose  periods  are  one-third,  one-fourth,  etc.,  that  of  the  funda- 
mental. The  general  condition  for  the  maintenance  of  stand- 
ing waves  in  an  open  pipe  is  that  there  shall  be  the  middle  of 
a  ventral  segment  at  each  end,  and  that  the  distance  between 
a  node  and  the  middle  of  a  ventral  segment,  or  one-quarter 
the  length  of  the  wave,  shall  divide  into  the  length  of  the  pipe 
an  even  number  of  times. 

If  the  pipe,  instead  of  being  open,  is  closed  at  the  top,  re- 
flection will  occur  there  with  change  of  sign.  At  the  open  end, 
where  the  disturbance  originates,  the  reflection  will  still  occur 
without  change  of  sign.  At  the  closed  end,  the  air  is  at  rest, 
and  therefore  that  point  in  the  column  is  a  node.  The  open 
end  is  at  the  middle  of  a  ventral  segment,  so  that  the  length 
of  the  pipe  is  one-quarter  the  length  of  the  wave  which  cor- 
responds to  its  fundamental  tone.  Or  more  fully,  if  an  up- 
ward displacement  leaves  the  open  end,  it  traverses  the  pipe, 
and  is  reflected  at  the  closed  end  as  a  downward  displacement. 
A  standing  wave  will  be  established  when  this  downward  dis- 
placement, on  reaching  the  bottom,  is  superposed  on  a  down- 
ward displacement  caused  by  the  body  setting  up  the  vibration. 
This  downward  displacement  will  be  set  up  at  a  time  later  by 
half  a  period  than  the  time  at  which  the  original  upward 
displacement  was  set  up,  so  that  the  time  taken  by  the  dis- 
turbance to  traverse  the  pipe  twice  is  equal  to  half  a  period. 

If  vibrations  occurring  twice  as  often  are  set  up  at  the 
open  end,  an  upward  displacement,  which  is  reflected  and 
returns  as  a  downward  displacement,  will  meet  another  up- 
ward displacement  at  the  open  end.  These  displacements, 
instead  of  reinforcing  each  other,  will  destroy  each  other.  A 
closed  pipe,  therefore,  will  not  give  the  octave  of  its  funda- 
mental. By  reasoning  generally  similar  to  that  already  em- 
ployed, it  may  be  shown  that  the  only  overtones  which  the 


SOUND.  133 

closed  pipe  can  emit  are  those  which  correspond  to  waves 
whose  lengths  are  the  uneven  fractional  parts  of  the  funda- 
mental wave  length.  The  general  condition  for  the  mainte- 
nance of  standing  waves  in  a  closed  pipe  is  that  there  shall  be 
a  node  at  the  closed  end,  the  middle  of  a  ventral  segment  at 
the  open  end,  and  that  the  distance  between  a  node  and  the 
middle  of  a  ventral  segment  shall  divide  into  the  length  of 
the  pipe  an  uneven  number  of  times. 

A  long  rod  of  wood,  glass,  or  metal,  if  clamped  in  the  mid- 
dle and  set  in  vibration  by  stroking  it  along  its  length,  will 
execute  vibrations  and  will  maintain  standing  waves  similar 
to  those  of  the  open  pipe.  As  in  the  case  of  the  air  in  the  pipe, 
the  length  of  the  wave  in  the  material  of  which  the  rod  is 
composed,  whose  period  is  that  of  the  emitted  tone,  is  twice 
the  length  of  the  rod.  An  arrangement  by  which  the  velocity 
of  sound  in  a  solid  may  be  determined,  depending  upon  this 
fact,  was  invented  by  Kundt.  It  consists  of  a  rod  of  the 
material  under  investigation,  clamped  in  the  middle,  and  with 
one  end  inserted  lightly  through  a  cork,  which  closes  one  end 
of  a  glass  tube.  "Fine  light  powder  is  scattered  within  the 
tube,  and  its  other  end  is  closed  with  a  loosely  fitting  cork, 
that  can  be  moved  up  and  down  in  it.  When  the  rod  is  made 
to  vibrate  longitudinally,  vibrations  of  the  same  period  are 
impressed  vipon  the  column  of  air  in  the  tube,  and  by  setting 
the  movable  cork  at  the  right  place,  they  may  be  made  to  set 
up  standing  waves  in  the  tube.  These  waves  stir  up  the 
powder  which  has  been  scattered  in  the  tube,  and  arrange  it 
in  a  regular  pattern,  from  which  the  positions  of  the  nodes 
may  be  determined,  and  so  the  length  of  the  wave  in  air, 
whose  period  is  that  of  the  tone  emitted  by  the  vibrating  rod. 
Since  the  length  of  the  corresponding  wave  in  the  rod  is  twice 
the  length  of  the  rod,  the  ratio  between  twice  the  length  of 
the  rod  and  the  length  of  the  wave  in  air  is  the  ratio  of  the 
velocity  of  sound  in  the  rod  to  the  velocity  of  sound  in  air. 
If  some  gas  other  than  air  is  used  in  the  tube,  the  wave  length 
determined  in  it,  compared  with  the  wave  length  produced  in 
air  by  similar  vibrations  of  the  rod,  affords  a  means  of  deter- 


134 

mining  the  velocity  of  sound  in  the  gas  and  so  of  testing  New- 
ton's formula. 

If  a  rod  is  clamped  at  one  end,  it  may  be  made  to  execute 
transverse  vibrations,  which  will,  if  sufficiently  rapid,  give 
rise  to  a  musical  tone.  When  the  rod  is  emitting  its  funda- 
mental tone,  {here  is  a  node  at  the  end  at  which  it  is  fixed. 
The  first  harmonic  which  is  developed  in  such  a  rod  is  about 
three  octaves  higher  than  the  fundamental. 

When  a  rod  which  is  free  at  both  ends  vibrates  trans- 
versely, it  develops  two  nodes,  which  are  distant  from  the  ends 
about  two-ninths  the  length  of  the  rod.  The  rod  may  be  sup- 
ported at  the  points  where  these  nodes  occur  without  inter- 
fering with  its  vibrations.  If  the  rod  is  bent  in  the  middle, 
the  two  nodes  approach  each  other,  until,  when  the  two 
halves  are  parallel,  the  nodes  are  very  near  together.  The 
tuning-fork  is  an  example  of  a  rod  bent  in  this  manner.  As 
the  ends  of  the  fork  swing  toward  each  other,  the  middle  of 
the  fork,  to  which  is  attached  the  stem  or  handle,  moves  down- 
ward. As  the  ends  move  apart,  the  middle  of  the  fork  moves 
upward.  Thus,  if  the  fork  is  sounding,  and  the  handle  is 
brought  down  on  the  table,  it  will  tap  the  table  top  at  regular 
intervals.  The  vibrations  thus  imparted  to  the  table  are  com- 
municated from  it  to  the  air,  and  thus  the  volume  of  sound 
emitted  by  the  fork  is  considerably  increased. 

Standing  waves  may  also  be  developed  in  plates  of  metal  or 
glass,  which  are  firmly  clamped  at  some  one  point  and  are 
bowed  or  stroked  at  some  point  on  the  edge.  The  nodes  of  the 
waves  thus  prodxiced  are  detected  by  sprinkling  a  little  sand 
over  the  plate,  which  is  thrown  away  from  the  vibrating  parts 
of  the  plate  and  collects  at  the  nodes.  This  method  of  obser- 
vation was  introduced  by  Chladni,  and  the  figures  obtained 
are  known  as  Chladni's  figures. 

94.  Quality  of  Sounds. — When  two  musical  instruments 
are  sounding  the  same  tone,  the  sounds  which  they  emit,  how- 
ever exactly  they  may  agree  in  pitch,  differ  entirely  in  quality. 
We  may  trace  this  difference  in  quality  to  the  other  tones  be- 
side the  fundamental  which  the  bodies  are  producing.  To  take 


SOUND.  135 

the  simplest  case,  tones  of  the  same  pitch  coming  from  an  open 
and  a  closed  organ  pipe  are  perceptibly  different  in  quality, 
the  tone  from  the  open  pipe  being  fuller  and  richer  than  the 
other.  This  difference  is  due  to  the  circumstance  already  dis- 
cussed, that  the  column  of  air  in  the  open  pipe  maintains 
standing  waves  corresponding  to  all  the  overtones  of  the 
fundamental,  while  that  in  the  closed  pipe  maintains  only 
those  corresponding  to  the  uneven  overtones.  When  a  pipe  is 
sounded  these  overtones  are  emitted  as  well  as  the  funda- 
mental, and  the  quality  of  the  tone  is  determined  by  their 
presence  or  absence  in  the  sound  and  by  their  relative  in- 
tensities. In  the  case  of  strings  all  overtones  may  be  present, 
and  further  additional  tones  occur,  which  are  produced  by 
very  rapid  vibrations  which  differ  considerably  with  the 
material  of  which  the  string  is  made.  It  is  in  general  true  of 
all  wind  instruments  also  that  the  tones  emitted  by  them  are 
distinguished  as  respects  their  quality  not  only  by  the  relative 
intensity  of  their  various  overtones,  but  also  by  characteristic 
tones,  or  even  by  characteristic  noises,  which  depend  upon  the 
material  and  the  construction  of  the  tube  in  which  the  air 
column  is  contained,  and  also  upon  the  way  in  which  the 
original  vibrations  are  produced. 

As  a  means  of  observing  the  presence  of  an  overtone  we 
use  an  instrument  designed  by  Koenig,  called  the  manometric 
capsule.  This  consists  of  a  small  box  divided  into  two  cham- 
bers by  a  thin  flexible  membrane.  One  of  these  chambers  is 
kept  filled  with  illuminating  gas,  which  is  burned  in  a  small 
jet  at  the  end  of  a  tube  projecting  from  the  chamber.  The 
other  chamber  is  connected  by  a  tube  to  the  source  of  the 
sound.  When  the  sound  is  given  forth,  the  gas  jet  is  viewed 
in  a  revolving  mirror.  The  vibrations  of  the  membrane  due 
to  the  sound  produce  changes  in  the  height  of  the  flame,  and 
the  band  of  light  which  is  seen  in  the  mirror  appears  serrated 
at  the  top.  If  the  tube  of  the  manometric  capsule  is  intro- 
duced into  the  side  of  an  organ  pipe,  at  a  place  where  a  node 
exists,  the  alternations  of  rarefaction  and  condensation  which 
occur  at  an  node  will  be  demonstrated  by  a  marked  serration 


136  SOUND. 

of  the  band  of  light.  If,  on  the  other  hand,  the  tube  is  intro- 
duced at  the  middle  of  a  ventral  segment,  where  there  are  no 
changes  of  density,  the  serrations  of  the  band  will  not  appear. 

The  overtones  of  pipes  may  be  produced  with  but  little 
admixture  of  the  lower  tones  by  changing  the  intensity  of  the 
blowing  and  often  by  a  manipulation  of  the  mouth  piece.  The 
existence  of  the  nodes  and  ventral  segments  corresponding  to 
the  overtones  may  be  demonstrated  by  the  manometric  capsule. 

By  touching  a  string  at  the  middle  and  bowing  it,  it  can 
be  made  to  emit  the  octave  of  its  fundamental.  When  it  is 
sounding  the  octave,  a  light  rider  of  paper  may  be  placed  at 
the  middle  point  and  will  remain  there  undisturbed,  whereas 
a  rider  anywhere  else  on  the  string  will  be  thrown  off.  Higher 
overtones  may  be  produced  in  a  similar  way  by  touching  the 
string  at  other  points,  and  the  existence  of  the  nodes  and 
ventral  segments  corresponding  to  them  may  also  be  demon- 
strated by  the  use  of  riders. 

The  most  complete  way  by  which  to  study  a  composite  tone 
is  by  the  use  of  resonators.  The  resonator  is  a  hollow  metallic 
sphere  with  a  circular  opening  on  one  side,  and  on  the  other 
a  short  projecting  tube,  to  which  the  ear  may  be  placed,  or  on 
which  the  tube  of  the  manometric  capsule  may  be  attached. 
The  air  within  the  sphere,  considered  as  a  sounding  body,  has 
a  fundamental  mode  of  vibration  and  but  very  few  and  feeble 
harmonic  vibrations.  If  this  resonator  is  in  the  presence  of  a 
body  which  is  emitting  the  fundamental  tone  of  the  resonator, 
the  air  in  it  will  be  set  in  vibration,  and  its  vibration  will  be 
indicated  by  the  manometric  capsule.  A  series  of  such 
resonators  may  be  made,  tuned  to  the  different  tones  of  the 
scale,  or  to  the  successive  overtones  of  a  fundamental  tone,  and 
can  then  be  used  to  investigate  the  different  tones  emitted  at 
the  same  time  by  a  sounding  body.  It  was  by  the  use  of  such  an 
instrument,  called  an  analyser,  that  Helmholtz  demonstrated 
the  relation  between  the  quality  of  a  tone  and  the  overtones 
present  in  it. 

95.  Beats  and  Resultant  Tones. — When  two  tuning  forks, 
or  two  pipes,  which  have  nearly  the  same  pitch,  are  sounded 


SOUND.  137 

together,  the  sound  heard  varies  in  intensity  periodically.  The 
complete  change  of  intensity  which  occurs,  from  the  greatest 
intensity  heard  through  the  least  to  the  greatest  again,  is 
called  a  beat.  Beats  may  be  explained  in  the  following  way: 
The  two  sounding  bodies  are  vibrating  in  periods  which  are 
nearly,  but  not  quite  the  same.  At  a  certain  instant  they  are 
moving  in  such  a  way  as  to  affect  the  air  around  them  in  a 
similar  manner.  The  disturbance  in  the  air  is  then  twice  as 
great  as  that  which  either  of  them  would  produce  if  it  were 
sounding  alone.  As  they  continue  to  vibrate,  their  motions 
become  less  and  less  similar,  until,  after  a  certain  time  has 
elapsed,  they  are  exactly  dissimilar  and  are  affecting  the  air 
around  them  oppositely.  The  sound  then  has  its  least  in- 
tensity. From  that  time  on,  the  sound  will  gradually  increase 
in  intensity,  until  one  of  the  bodies  has  gained  a  whole  vibra- 
tion on  the  other,  when  the  sound  will  again  have  its  greatest 
intensity.  Since  the  time  between  two  instants  of  greatest 
intensity  is  that  required  for  one  body  to  execute  one  vibra- 
tion more  than  the  other,  the  number  of  beats  heard  in  one 
second  will  be  equal  to  the  difference  between  the  vibration 
numbers  of  the  two  bodies. 

Koenig  has  shown  that  beats  may  also  be  heard  when  two 
bodies  are  sounding  together  whose  pitch  is  very  different, 
provided  that  the  two  tones  are  nearly  an  octave  apart,  or 
are  nearly  in  some  other  harmonious  relation. 

When  two  tones  are  sounded  together  whose  vibration 
numbers  differ  considerably,  a  third  tone  is  often  heard,  called 
the  resultant  tone.  The  vibration  number  of  this  tone  is 
equal  to  the  difference  between  the  vibration  numbers  of  the 
other  tones.  Koenig's  observations  make  it  probable  that 
resultant  tones  are  caused  by  the  regular  beats  that  occur 
when  the  two  primary  tones  are  sounded.  Other  resultant 
tones  are  heard  when  two  primary  tones  are  sounded  strongly, 
which  Helmholtz  explained  on  the  hypothesis  that  the  move- 
ments of  the  air  set  up  by  the  sounding  bodies  are  so  great 
that  the  simple  law  of  elasticity  is  not  longer  applicable. 


138  SOUND. 

90.  Harmony. — It  is  interesting  to  observe,  that  those 
tones  whose  combination  is  most  pleasing  and  gives  most  fully 
the  sense  of  harmony,  are  those  whose  vibration  numbers  are 
in  the  simplest  proportion  to  each  other.  Thus,  next  to  the 
combination  of  two  tones  which  are  in  unison,  or  whose  vibra- 
tion numbers  are  the  same,  the  most  perfect  harmony  is 
obtained  when  the  octave  is  sounded  with  the  fundamental; 
that  is,  by  the  combination  of  two  tones  wThose  vibration  num- 
bers are  as  2  to  1.  The  next  most  harmonious  co'mbmation  is 
that  of  the  fifth  with  the  fundamental,  whose  vibration  num- 
bers are  as  3  to  2.  The  combination  of  the  fourth  with  the 
fundamental,  whose  vibration  numbers  are  as  4  to  3,  and  of 
the  third  with  the  fundamental,  whose  vibration  numbers  are 
as  5  to  4,  are  also  harmonious.  On  the  other  hand,  the  com- 
bination of  the  seventh  with  the  octave,  whose  vibration  num- 
bers are  as  15  to  16,  is  not  harmonious.  In  this  case  beats  are 
heard.  It  seems  probable  that  discordant  combinations  or 
discords  occur  when  beats  are  produced  which  lie  between 
certain  limits.  Tones  which  produce  fewer  beats  than  10  per 
second,  or  more  beats  than  70  per  second,  are  not  discordant, 
but  if  the  number  of  beats  produced  lies  between  those  limits, 
the  tones  are  discordant. 

97.  Absolute  S umber  of  Vibrations. — So  far  we  have  con- 
sidered mainly  the  relative  number  of  vibrations  executed  by 
a  sounding  body,  or  producing  a  sound,  in  terms  of  the  vibra- 
tions of  some  body  taken  as  a  standard.  It  is,  however,  a 
matter  of  interest  to  determine  the  absolute  number  of  vibra- 
tions which  corresponds  to  a  certain  pitch.  The  determination 
of  this  number  may  be  made  in  several  ways. 

It  has  already  been  explained  how  Mersenne,  by  using  a 
string  so  long  that  its  vibrations  could  be  counted,  and  com- 
paring its  length  with  that  of  a  shorter  string  under  the  same 
tension  and  emitting  a  standard  tone,  was  able  to  determine 
the  number  of  vibrations  corresponding  to  that  tone. 

Sauveur  used  the  beats  produced  by  two  organ  pipes,  whose 
vibration  numbers  were  in  a  known  ratio,  to  determine  their 
absolute  vibration  numbers.  The  pipes  which  he  used  gave  the 


SOUND.  139 

fundamental  and  the  seventh  of  the  next  lower  octave.  Their 
vibration  numbers  were  therefore  in  the  ratio  of  16  to  15.  In 
the  sound  which  was  heard  when  they  were  sounded  together 
there  were  six  beats  per  second,  so  that  the  difference  of  their 
vibration  numbers  was  6.  From  these  two  relations  it  follows 
that  the  vibration  numbers  of  the  two  tones  were  90  and  96. 

Savart  used  a  toothed  wheel,  which  was  rotated  by  a 
mechanism,  so  constructed  that  its  rate  of  rotation  could  be 
controlled  and  that  the  number  of  rotations  per  second  could 
be  counted.  When  the  wheel  was  in  rotation  and  a  flexible 
card  brought  up  against  the  teeth,  a  musical  tone  was  emitted, 
and  by  regulation  of  the  rate  of  rotation  this  tone  could  be 
brought  in  harmony  with  that  of  a  standard  pipe  or  fork.  The 
number,  of  vibrations  in  the  standard  tone  was  then  deter- 
mined from  the  number  of  rotations  made  by  the  wheel  in  a 
second,  and  the  number  of  teeth  on  the  wheel. 

Duhamel  set  a  light  pointer  on  the  end  of  a  tuning  fork, 
and  arranged  it  so  that  the  fork  when  sounding  was  carried 
along  over  a  plate  of  glass  covered  with  a  thin  coating  of  lamp 
black.  The  pointer  traced  out  a  sinuous  line  on  the  blackened 
surface,  and  the  number  of  vibrations  executed  by  the  fork 
was  equal  to  the  number  of  sinuosities  in  this  line.  By  means 
of  an  additional  mechanism,  a  time-keeper  was  made  to  record 
equal  intervals  of  time  on  the  same  blackened  surface,  and  by 
counting  the  number  of  sinuosities  lying  between  two  marks 
made  by  the  time-keeper,  the  vibration  number  of  the  fork  was 
determined. 

The  vibration  number  of  a  tone  may  also  be  determined  by 
means  of  the  siren.  The  siren  is  an  instrument  which  pro- 
duces a  sound  by  means  of  the  alternate  emission  and  suppres- 
sion of  puffs  of  air  at  regular  intervals.  It  consists  of  a  box 
or  air  chest  with  a  flat  top  in  which  are  pierced  a  number  of 
holes  set  uniformly  .around  a  circle.  A  flat  disk  is  mounted 
on  an  axle,  which  turns  in  a  support  set  at  the  centre  of  this . 
circle,  and  the  disk  is  set  as  near  the  top  of  the  box  as  it  can 
be  without  touching  it.  A  set  of  holes  is  made  in  the  d4'*, 
which  correspond  in  number  and  position  to  those  in  the  iid 


140  SOUND. 

of  the  box.  Sometimes,  by  setting  the  holes  obliquely,  the  air 
which  is  forced  out  of  the  box  may  be  made  to  turn  the  disk, 
but  it  is  best  lo  turn  the  disk  by  some  outside  mechanism.  A 
counter  is  provided  by  means  of  which  the  rate  of  rotation  of 
the  disk  may  be  determined.  When  the  disk  is  turned  and  air 
is  forced  into  the  box,  a  puff  of  air  "comes  out  of  each  hole 
when  the  holes  in  the  disk  stand  directly  over  those  in  the 
box.  When  the  disk  turns  on  so  that  the  holes  in  the  box  are 
covered,  the  air  is  shut  off.  The  number  of  puffs  emitted  dur- 
ing one  rotation  of  the  disk  is  equal  to  the  number  of  holes 
in  the  disk.  To  determine  the  vibration  number  of  a  given 
tone,  the  disk  is  turned  until  the  tone  emitted  by  the  instru- 
ment is  in  unison  with  the  one  whose  vibration  number  is 
desired.  When  this  condition  is  obtained,  the  rotation,  of  the 
disk  is  maintained  uniform,  and  the  number  of  its  rotations 
per  second  is  determined  by  the  counter.  From  this  number 
and  the  number  of  holes  in  the  disk,  the  vibration  number  of 
the  tone  may  be  obtained. 


141 


HEAT. 

98.  Sensation  of  Heat. — The  sensation  by  which  we  dis- 
tinguish between  hot  and  cold  bodies  is,  and  must  always  have 
been,  a  perfectly  familiar  one.  The  sense  by  which  we  per- 
ceive it  may  be  called  the  temperature  sense.  It  does  not  seem 
to  be  the  same  as  the  sense  of  touch,  by  which  we  distinguish 
the  forms  of  bodies.  The  sensations  given  by  it  depend  upon 
so  many  conditions  that  they  are  utterly  untrustworthy  as  a 
measure  of  the  temperature  of  the  body  which  we  examine  by 
it.  Thus  the  same  mass  of  water  will  appear  to  the  hand  either 
warm  or  cold,  according  as  the  hand  has  previously  been  im- 
mersed in  very  cold  or  in  very  hot  water;  and  two  bodies  of 
different  material,  like  wood  and  iron,  which  have  been  ex- 
posed to  the  same  conditions,  and  which  all  physical  tests  show 
us  must  be  at  the  same  temperature,  will  not  appear  equally 
hot  or  cold  when  tested  by  the  hand.  Men  have  always  been 
accustomed  to  speak  of  the  cause  of  the  sensation  felt,  when 
we  touch  a  hot  body,  as  heat,  and  to  assign  the  different  sensa- 
tions given  us  by  the  body  in  different  circumstances  to  the 
presence  in  it  of  more  or  less  heat.  Similarly  the  sensation 
experienced  when  we  touch  a  cold  body  has  often  been  assigned 
to  the  presence  in  the  body  of  something  called  cold.  But, 
after  all,  the  distinction  between  the  sensation  itself  and  its 
cause  was  not  very  sharply  drawn,  and  a  great  deal  of  con- 
fusion exists  in  the  early  work  on  the  subject  beween  the  two 
ideas  of  temperature  and  quantity  of  heat.  Our  sensations 
immediately  recognize  temperature  and  changes  of  tempera- 
ture, and  attention  was  first  turned  to  the  study  of  those 
physical  relations  of  bodies  which  are  connected  with  their 
temperature. 

Our  object  in  the  study  of  heat  is  to  trace  the  sensation  of 
heat  to  its  origin  in  material  bodies,  and  to  explain  the 
various  phenomena  exhibited  by  bodies  in  connection  with 
heat  in  terms  of  matter  and  the  motions  of  matter. 


142 


99.  Thermometers. — Our  sensations  being  so  uncertain  in 
their  estimate  of  the  temperature  of  bodies,  it  is  of  first  im- 
portance to  obtain  an  instrument  which  will  indicate  tem- 
perature. Such  an  instrument  is  called  a  thermometer. 

The  first  thermometer  was  constructed  by  Galileo.  It  con- 
sisted of  a  small  glass  globe  fitted  with  a  long  tube.  The  open 
end  of  the  tube  was  inserted  in  water,  and  the  globe  was  heated 
until  some  of  the  air  was  expelled  from  it.  When  it  was 
allowed  to  cool,  the  pressure  of  the  external  air  was  greater 
than  that  of  the  air  in  the  globe,  and  so  a  column  of  water  was 
forced  up  the  tube,  until  equilibrium  was  established.  The 
air  thus  shut  off  in  the  globe  formed  what  we  may  call  the 
thermometric  substance,  and  the  top  of  the  water  column  de- 
termined its  volume.  In  using  this  instrument,  as  in  using 
any  thermometer,  it  was  assumed,  as  a  fundamental  fact  of 
experience,  that  when  two  bodies,  whose  temperatures  are 
different,  are  brought  into  each  others  presence,  their  tem- 
peratures will  finally  become  equal.  This  equalization  of  tem- 
perature is  brought  about  by  a  lowering  of  the  temperature  of 
the  hotter  body,  and  a  raising  of  the  temperature  of  the 
colder  body.  It  was  also  assumed  that  a  change  in  the  tem- 
perature of  a  body  is  accompanied  by  a  change  in  its  volume, 
and  in  particular  that  the  volume  of  air  increases  as  its  tem- 
perature rises.  Starting  with  these  fundamental  principles, 
the  use  of  the  thermometer  as  constructed  by  Galileo  is  evi- 
dent. When  it  is  kept  in  a  room  at  the  ordinary  temperature, 
the  top  of  the  water  column  will  stand  at  a  certain  point  in 
the  tube,  which  may  be  marked.  If  it  is  then  transferred  to  a 
hotter  room,  or  if  a  hot  body  is  placed  around  the  globe,  the 
equalization  of  temperature  already  described  will  occur,  and 
the  volume  of  the  air  in  the  instrument  will  increase.  The 
distance  which  the  top  of  the  water  column  is  forced  down 
the  tube  is  a  measure  of  the  change  of  temperature.  For 
various  reasons,  the  principal  one  of  which  is  the  irregularity 
introduced  by  changes  in  the  external  atmospheric  pressure, 
Galileo's  thermometer  will  not  give  consistent  indications  of 


HEAT.  143 

temperature.  It  was  very  goon  superseded  by  instruments 
constructed  on  a  different  plan. 

Stimulated,  no  doubt,  by  this  invention  of  Galileo,  a  body 
of  physicists,  resident  at  Florence  and  united  in  a  club  called 
the  Accademia  del  Cimento,  undertook  the  construction  of 
thermometers  which  should  furnish  satisfactory  measures  of 
temperature.  The  instrument  they  made  was  similar  in  form 
to  the  ordinary  thermometers  now  in  use.  That  is,  it  con- 
sisted of  a  glass  bulb  joined  to  a  graduated  tube,  the  bulb  and 
part  of  the  tube  being  filled  with  a  liquid.  The  peculiarity  of 
their  instrument  consisted  in  this,  that  they  attempted  to 
graduate  the  tube  in  such  a  way  that  the  volume  between  two 
marks  of  graduation  was  a  fixed  fractional  part,  generally 
one-thousandth,  of  the  volume  of  the  bulb.  The  instrument 
thus  made  was  filled  with  alcohol,  so  that  the  top  of  the 
column  stood  opposite  one  of  the  marks  on  the  scale  when  the 
instrument  was  exposed  to  some  standard  temperature.  The 
temperature  chosen  as  standard  was  that  of  the  air  during  the 
first  light  frosts  at  the  beginning  of  the  winter.  Their  first 
instruments  were  open  at  the  top,  but  as  this  interfered  with 
their  permanence  and  with  their  transportability,  they  were 
afterwards  closed.  This  plan  of  constructing  thermometers 
did  not  succeed  in  furnishing  instruments  which  would  give 
similar  indications  at  different  temperatures,  and  notwith- 
standing the  partial  success  which  was  obtained  much  later 
by  Keaumur  in  the  construction  of  thermometers  on  the  same 
plan,  it  has  long  ago  been  abandoned. 

The  method  of  graduating  thermometers  which  is  now  uni- 
versally used  was  describc-d  by  Dalence  in  1688.  and  an  instru- 
ment made  on  that  plan  was  constructed  by  Newton.  In  it  no 
attempt  is  made  to  establish  any  fixed  relation  between  the 
volumes  indicated  by  the  graduation  and  the  volume  of  the 
bulb.  Only  so  much  choice  is  exercised  of  the  relative  volumes 
of  the  tube  and  the  bulb  as  will  ensure  that  the  thermometer 
can  be  used  throughout  the  temperature  range  for  which  it  is 
intended.  Newton  used  linseed  oil  as  the  thermometric  sub- 
stance. To  graduate  the  instrument,  he  placed  it  first  in  a 


144  HEAT. 

mixture  of  ice  and  water,  the  temperature  of  which  waa 
known  to  remain  constant.  He  made  a  mark  on  the  tube  at 
the  point  indicated  by  the  top  of  the  liquid  column,  after  the 
instrument  had  stood  in  the  ice  for  some  time,  and  showed  no 
signs  of  any  further  change.  This  point  indicated  one  stan- 
dard temperature.  He  then  placed  the  bulb  of  the  instrument 
under  his  arm  pit,  and  after  the  column  again  became  station- 
ary, he  made  another  mark  on  the  tube  opposite  the  end  of 
the  column.  The  temperature  of  the  human  body  was  known 
to  be  very  nearly  constant,  and  the  temperature  thus  obtained 
was  therefore  taken  by  Newton  as  a  second  standard  temper- 
ature. The  distance  between  the  two  standard  marks  thus 
obtained  was  divided  into  twelve  equal  parts,  and  the  gradu- 
ation thus  established  was  extended  above  and  below  the 
standard  marks.  Newton  did  not  choose  the  best  thermo- 
metric  substance  that  can  be  used,  or  the  most  suitable  stan- 
dard temperatures,  but  the  method  which  he  employed  was 
essentially  correct.  It  is  easy  to  see  that  if  two  instruments, 
in  which  the  same  thermometric  substance  is  used,  are  gradu- 
ated in  this  manner,  they  will  not  only  indicate  the  standard 
temperatures  under  the  same  conditions,  but  will  agree  in 
their  indications,  to  whatever  temperature  they  are  exposed. 
The  use  of  mercury  as  a  thermometric  substance  was  intro- 
duced by  Fahrenheit,  to  Avhom  we  owe  the  first  thermometers 
which  compare  with  those  now  made  in  the  accuracy  of  their 
indications.  Fahrenheit  used  the  temperature  of  melting  ice 
as  one  of  the  standard  temperatures,  and  probably  the  tem- 
perature of  boiling  water  under  a  standard  pressure  as  the 
other.  It  was  subsequently  discovered  that  the  temperature 
of  boiling  wrater  not  only  depends  upon  the  atmospheric  pres- 
sure, but  also  upon  the  material  of  the  vessel  in  which  the 
water  is  boiling.  Cavendish  therefore  proposed  to  use  as  the 
second  standard  temperature  the  temperature  of  steam  over 
boiling  water,  which  -he  showed  to  be  independent  of  the 
material  of  the  vessel.  It  depends  upon  the  atmospheric  pres- 
sure in  a  way  which  has  been  carefully  determined  by  experi- 
ment, so  that  a  correction  can  be  made  to  standard  pressure. 


HEAT.  145 

These  two  temperatures,  the  temperature  of  melting  ice,  and 
the  temperature  of  steam  over  boiling  wfeter  at  a  standard 
pressure,  are  those  now  universally  adopted  as  the  two  stan- 
dard temperatures. 

Fahrenheit  assigned  the  number  32  to  the  first  standard 
mark,  and  the  number  212  to  the  second  standard  mark.  He 
therefore  divided  the  distance  between  the  two  marks,  or  as 
we  may  call  them,  the  freezing  point  and  the  boiling  point, 
into  180  parts  or  degrees.  When  this  scale  is  extended  below 
the  freezing  point,  the  zero  marks  a  temperature  which  is 
very  nearly  that  of  a  mixture  of  pounded  ice  and  salt. 

A  change  in  the  number  of  degrees  between  the  freezing 
point  and  the  boiling  point  was  recommended  by  Linnaeus  and 
carried  out  by  Celsius.  In  the  thermometers  of  Celsius  there 
were  100  degrees  between  the  two  standard  temperatures. 
The  temperature  of  the  freezing  point  was  marked  0,  so  that 
the  temperature  of  the  boiling  point  was  100.  The  scale  thus 
constructed  is  now  generally  called  the  Centigrade  scale.  It 
is  the  one  universally  used  in  physical  investigations,  though 
the  Fahrenheit  scale  is  still  used  in  England  and  America  by 
meteorologists. 

It  should  be  observed  that  the  scale  of  temperature  which 
has  been  adopted  is  a  purely  arbitrary  one.  The  zero  is  an 
arbitrarily  chosen  temperature,  and  the  change  of  temperature 
which  is  called  a  degree  is  determined  by  an  arbitrarily  chosen 
change  of  volume  of  a  standard  substance.  There  is  nothing 
which  tells  us  that  the  change  of  temperature  which  causes 
this  standard  change  of  volume  in  one  part  of  the  scale  is  the 
same  as  the  change  of  temperature  which  will  cause  the  same 
change  of  volume  in  another  part  of  the  scale,  it  being  under- 
stood that  by  change  of  temperature  in  this  statement  is 
meant  a  change  measured  by  a  change  in  the  fundamental 
physical  condition  of  the  body,  which  is  the  true  measure  of 
its  temperature.  This  thermometric  scale,  therefore,  does  not 
furnish  an  absolute  measure  of  temperature,  and  strictly 
speaking  only  indicates  differences  of  temperature. 


146  HEAT. 

Many  other  arrangements  are  used  for  the  measurement  of 
temperatures  and  temperature  differences.  They  depend  for 
their  operation  upon  some  relation  between  some  physical 
property  of  a  body  which  can  be  measured  and  its  temperature. 
Their  indications  are  usually  compared  with  those  of  a  ther- 
mometer, and  are  thus  expressed  in  degrees. 

100.  Melting  Temperatures,  etc. — Even  with  their  imper- 
fect thermometers  the  early  observers  made  several  important 
discoveries  in  connection  with  the  subject  of  temperature.  The 
Accademia  del  Cimento  tried  the  experiment  of  immersing  & 
vessel  filled  with  ice  in  a  large  mass  of  hot  water,  and  observ- 
ing the  temperature  of  the  ice  with  one  of  their  thermometers. 
They  expected  to  find  that  the  temperature  of  the  ice  would 
fall,  but  found,  in  fact,  that  it  remained  constant  until  most 
of  the  ice  was  melted.  By  trial  with  other  masses  of  ice, 
they  found  that  ice  always  melts  at  appreciably  the  same  tem- 
perature. This  temperature  is  called  the  melting  point  of  ice. 
Their  observation  was  frequently  confirmed  by  other  observers, 
and  it  was  shown  further  that  many  other  bodies  possess 
definite  melting  points,  which  are  characteristic  of  the  bodies. 
By  selecting  a  set  of  such  bodies,  melting  at  different  tem- 
peratures, a  series  of  definite  temperatures  may  be  determined, 
which  are  independent  of  the  construction  of  any  particular 
instrument,  and  so  far  as  we  know,  will  be  the  same  every- 
where. The  first  scale  of  this  sort  was  established  by  Newton. 

Hooke  discovered  that,  when  water  is  boiled,  its  tempera- 
ture is  always  approximately  constant.  This  temperature, 
called  the  boiling  point  of  water,  is  not  so  independent  of 
external  conditions  as  the  freezing  point  is.  In  particular,  it 
depends  upon  the  pressure  in  the  vessel  in  which  the  water  is 
boiled.  This  had  been  proved  some  time  before  Hooke's  dis- 
covery by  Boyle,  who  placed  a  vessel  of  water  which  was  hot, 
but  not  boiling,  in  the  receiver  of  his  air  pump,  and  exhausted 
the  air  from  around  it.  When  the  exhaustion  had  reached  a 
certain  point,  the  water  began  to  boil,  and  it  could  be  made  to 
boil  again  and  again  by  exhausting  the  receiver  still  farther, 
although  it  was  continually  cooling.  On  the  other  hand,  if 


HKAT.  147 

water  is  enclosed  in  a  tight  vessel,  like  a  boiler,  under  high 
pressure,  it  will  not  boil  unless  its  temperature  is  raised  far 
above  its  boiling  point  in  an  open,  vessel. 

When  water  is  boiled  in  an  open  vessel,  it  is  under  atmos- 
pheric pressure,  and  this  changes  from  time  to  time,  so  that 
the  boiling  point  of  water,  when  tested  by  an  accurate  ther- 
mometer, will  not  appear  to  be  the  same  at  all  times.  It  will 
be  the  same,  however,  if  examined  at  times  when  the  atmos- 
pheric pressure  is  the  same.  In  order  to  use  the  boiling  point 
as  a  standard  temperature,  the  standard  pressure  of  one  atmos- 
phere, or  of  700  millimetres  of  mercury,  has  been  selected  as 
the  one  at  which  the  boiling  point  shall  be  standard. 

101.  Freezing  Mixtures. — The  Accademia  del  Cimento  dis- 
covered several  pairs  of  substances,  which,  when  mixed  with 
each  other,  would  produce  very  low  temperatures.     Such  mix- 
tures were  called  freezing  mixtures,  because  the  temperatures 
which  they  produced  were  so  low  that  other  bodies  could  be 
frozen  by  means  of  them.     The  mixture  of  ice  and  salt  is  a 
familiar  example.     If  a  quantity  of  broken  ice,  at  a  tempera- 
ture below  the  freezing  point,  while  it  is  therefore  a  dry  solid, 
is  mixed  with  salt,  the  temperature  of  the  mixture  falls  until 
it  reaches  0°  Fahrenheit,  or  ahout  — 1H°  (.'enuirrade.     Of  course 
it  cools  the  bodies  around  it,  and  thus  may  be  used  to  freeze  a 
liquid  brought  in  contact  with  it. 

It  was  noticed  by  Boyle,  who  devoted  considerable  atten- 
tion to  freezing  mixtures,  that  the  solid  bodies  which  were 
brought  together  in  the  mixture  always  melted,  or,  that  if  they 
were  melting  already,  they  melted  faster  after  being  mixed. 
Thus,  in  the  mixture  of  dry  ice  and  salt  already  described,  both 
ice  and  salt  melt  at  a  temperature  below  the  melting  point  ot 
either  one  of  them. 

102.  Freezing  Points. — When  water  is  exposed  in  a  vessel, 
for  a  sufficient  time,  to  a  temperature  which  is  below  its  melt- 
ing point,  it  will  gradually  freeze.     While  freezing,  its  tem- 
perature remains  constant  at  the  melting  point,  that  is,  the 
melting  and   freezing  temperatures  are  the  same.     This  fact 
was  not  recognized  by  the  earliest  observers,  because  of  the 


148  HKAT. 

way  in  whicli  the  temperatures  of  different  parts  of  a  mass  of 
water  differ  on  account  of  their  differences  in  density,  but  it 
was  easily  observed  when  small  quantities  of  water  were 
rapidly  frozen  by  means  of  freezing  mixtures.  The  same  gen- 
eral statement  holds  true  for  other  bodies  which  have  definite 
melting  points,  that  their  melting  points  and  freezing  points 
are  the  same. 

An  apparent  exception  to  this  rule  was  discovered  by 
Fahrenheit.  He  showed  that  if  a  small  quantity  of  water  was 
first  boiled,  so  as  to  expel  the  air  from  it,  and  then  allowed  to 
cool  slowly  in  a  smooth  glass  vessel,  its  temperature  might  fall 
several  degrees  below  the  freezing  point  without  its  freezing. 
The  water  in  this  condition  is  said  to  be  supercooled.  If 
supercooled  water  is  suddenly  agitated,  or  if  a  grain  of  sand, 
or  better  still,  a  crystal  of  ice,  is  dropped  in  it,  it  will  at  once 
begin  to  freeze.  Freezing  goes  on  in  this  case  much  more 
rapidly  than  when  it  begins  at  the  freezing  temperature.  At 
the  same  time,  the  temperature  rises  to  the  normal  freezing 
point.  Very  many  liquids  may  be  supercooled  in  a  similar 
way.  and  exhibit  similar  phenomena  when  they  freeze. 

103.  Change  of  Volume  on  Freezing. — When  a  liquid  freezes 
there  generally  occurs  a  rearrangement  of  its  parts,  such  that 
the  density  of  the  solid  formed  is  different  from  that  of  the 
liquid.     Galileo  noticed  that  this  is  true  in  the  case  01  water 
and  ice,  and  showed,  from  the  fact  that  ice  floats  in  water, 
that  the  density  of  the  ice  is  less  than  that  of  the  water.    The 
relative  density  of  ice  to  water  is  about  as  0.918  to  1.    Metals 
like  bismuth,  type  metal,  or  even  iron,  with  which  sharp  cast- 
ings can  be  made,  agree  with  water  in  having  the  density  of 
the  solid  state  less  than  that  of  the  liquid.    In  most  cases  the 
change  is  in  the  opposite  sense,  and  the  density  of  the  solid 
is  greater  than  that  of  the  liquid. 

104.  The    Temperature   of    Mixtures. — Taylor,    and    after- 
wards Riehmann,  tried  the  experiment  of  mixing  quantities  of 
water  together  whose  temperatures  were  different,  and  observ- 
ing the  resulting  temperature  of  the  mixture.     They  found 
that,  when  the  two  quantities  of  water  were  equal,  the  result- 


HEAT.  149 

ing  temperature  was  the  mean  of  the  original  temperatures. 
When  tne  quantities  of  water  were  not  equal,  this  was  not  the 
case.  The  resulting  temperature  was  found  to  be  given  by 
the  following  rule:  Multiply  the  mass  of  each  portion  of 
water  by  its  temperature,  add  the  products,  and  divide  the 
sum  by  the  sum  of  the  masses;  the  quotient  is  the  resulting 
temperature.  This  rule  is  known  as  Richmann's  rule. 

We  may  explain  tne  fact  embodied  in  Richmann's  rule,  if 
we  suppose  that  the  changes  of  temperature  which  occur  in 
the  two  masses  of  water  are  due  to  the  passage  of  heat  from 
one  to  the  other.  That  is,  we  suppose  that,  when  the  tem- 
perature of  water  rises,  it  is  because  of  the  entrance  into  the 
water  of  something  which  we  call  heat,  and  which  we  believe 
to  be  the  cause  of  temperature.  If  we  assume  that  the 
change  of  temperature  produced  by  the  mixture  of  the  two 
portions  of  water  is  due  to  the  passage  of  heat  from  the  hotter 
portion  into  the  colder,  we  find  that  Richmann's  rule  is  con- 
sistent with  this  assumption,  if  the  quantity  of  heat  which 
enters  a  body  is  proportional  to  the  mass  of  the  body  and  to 
the  rise  of  temperature  which  occurs.  For,  Richmann's  rule 
is  equivalent  to  the  statement  th'at  the  products  of  the  masses 
of  water  and  their  respective  changes  of  temperature  are 
equal. 

We  have  thus  reached  a  conception  of  heat  as  something 
which  may  enter  or  leave  a  body,  which  is  distributed  through- 
out its  mass,  and  which  determines  its  temperature.  We  have 
further  found  a  way  to  measure  it,  at  least  to  measure  so 
much  of  it  as  enters  or  leaves  a  body,  by  the  observation  of 
the  mass  of  the  body  and  of  its  change  of  temperature.  By 
selecting  a  particular  body  and  a  particular  change  of  tem- 
perature, we  may  define  a  unit  of  heat. 

A  unit  of  heat  which  is  frequently  used  in  physical  investi- 
gations is  called  the  calorie.  It  is  the  heat  which  will  raise 
the  temperature  of  a  kilogramme  of  pure  water  one  degree 
Centigrade.  As  recent  observation  has  shown  that  the  amount 
of  neat,  measured  in  energy  units,  which  will  raise  the  tem- 
perature of  a  kilogramme  of  water  one  degree,  is  slightly 


150  H  KAT. 

different  in  different  parts  of  the  scale,  it  is  necessary,  in  order 
to  give  greater  precision  to  this  definition,  to  specify  the  par- 
ticular degree  on  the  scale  through  which  the  temperature  of 
the  water  shall  be  raised.  The  degree  usually  chosen  is  that 
between  0°  and  1°  on  the  Centigrade  scale,  though  other  de- 
grees have  been  chosen.  It  is  often  convenient  to  use  a 
smaller  unit  of  heat  than  this,  and  we  accordingly  choose,  as 
another  unit,  the  heat  which  will  raise  the  temperature  of  a 
gramme  of  water  from  0°  to  1°  Centigrade.  This  unit  we  may 
call  the  gramme-degree. 

105.  Specific  Heat. — When  masses  of  two  different  sub- 
stances, whose  temperatures  are  different,  are  mixed,  the 
resulting  temperature  does  not  conform  to  Bichmann's  rule. 
It  was  discovered  by  Black  that  the  resulting  temperature 
may  always  be  found,  in  a  way  consistent  with  the  general 
conception  of  heat  which  we  fcave  adopted,  and  by  a  rule 
essentially  similar  to  Richmann's,  if  we  suppose  that  the 
amount  of  heat  required  to  raise  the  temperature  of  unit  mass 
of  a  particular  substance  one  'degree,  is  characteristic  or  spe- 
cific for  that  substance.  On  this  supposition,  the  quantity  of 
heat  which  a  mass  of  a  subst'ance  will  lose  when  its  tempera- 
ture falls  a  certain  number  of  degrees,  is  equal  to  the  specific 
heat  of  the  substance  multiplied  by  its  mass  and  by  its 
change  of  temperature.  When  portions  of  two  different  sub- 
stances are  mixed,  so  that  heat  passes  from  one  to  the  other, 
the  resulting  lemperature  is  found  by  a  rule  similar  to  Rich- 
mann's rule  if  we  replace  the  masses  in  Richmann's  rule  by 
the  products  of  the  masses  and  the  specific  heats. 

From  the  rule  that  has  just  been  given,  we  may  determine 
the  ratio  of  the  specific  heats  of  any  two  substances,  by  allow- 
ing known  masses  of  them,  whose  original  temperatures  differ 
from  each  other,  to  exchange  heat  until  they  reach  a  common 
temperature.  If,  in  all  our  experiments,  we  use  some  one  sub- 
stance as  standard,  we  may  determine  the  ratio  of  the  specific 
heats  of  other  substances  to  that  of  the  standard  substance, 
and  if  we  arbitrarily  assign  the  value  1  to  the  specific  heat  of 


HKAT.  151 

the  standard  substance,  the  ratios  thus  determined  may  be 
called  the  specific  heats  of  the  other  substances. 

In  the  determination  of  the  specific  heats  of  solids  and 
liquids,  the  substance  universally  chosen  as  the  standard  is 
water.  The  specific  heat  of  any  other  substance  is  therefore 
the  ratio  of  the  amount  of  heat  which  will  raise  the  tempera- 
ture of  a  mass  of  that  substance  one  degree  to  the  amount  of 
heat  which  will  raise  the  temperature  of  an  equal  mass  of 
water  one  degree.  If  the  mass  considered  in  this  definition  is 
a  kilogramme,  the  ratio  between  the  two  quantities  of  heat  is 
the  amount  of  heat,  measured  in  calories,  which  will  raise  the 
temperature  of  a  kilogramme  of  the  substance  one  degree.  We 
may  therefore  define  the  specific  heat  of  a  substance  as  the 
amount  of  heat,  measured  in  calories,  which  will  raise  the 
temperature  of  a  kilogramme  of  the  substance  one  degree. 
The  heat  capacity  of  a  body  is  equal  to  the  product  of  its  mass 
and  its  specific  heat. 

106.  Latent  Heat  or  Heat  of  Fusion. — In  the  investigation 
of  specific  heat,  it  was  found  by  Black  that  the  law  by  which 
the  resulting  temperature  of  a  mixture  is  determined  does  not 
hold  good  if  one  of  the  bodies  melts  when  it  is  brought  in  con- 
tact with  the  other.  In  any  such  case,  the  resulting  tempera- 
ture is  alway*s  lower  than  that  given  by  the  law.  Black  was 
therefore  led  to  consider  the  process  of  melting,  in  order  to 
ascertain  whether  heat  is  required  for  it.  In  one  of  his  ex- 
periments he  placed  two  similar  vessels,  one  containing  water, 
the  other,  an  equal  mass  of  ice,  on  the  top  of  a  stove.  The 
temperature  of  the  water  at  once  began  to  rise.  The  ice  in 
the  other  vessel  melted,  but  its  temperature,  and  that  of  the 
water  which  flowed  from  it,  remained  constant  until  melting 
was  complete.  Then  the  temperature  in  that  vessel  also  began 
to  rise,  and  rose  at  the  same  rate  as  in  the  other  vessel.  It 
is  plain  that  heat  must  have  been  entering  both  vessels  all  the 
time  and  at  the  same  rate,  and  since  no  evidence  of  its  having 
entered  the  ice  was  given  by  any  change  of  temperature,  it 
must  have  been  somehow  used  in  melting  the  ice.  In  another 
of  his  experiments,  Black  placed  a  mass  of  ice  in  a  mass  of 


152  HEAT. 

warm  water,  and  determined  the  temperature  which  resulted 
when  the  ice  was  melted.  On  the  supposition  that  the  water 
formed  by  the  melting  ice  mixed  with  the  warm  water  accord- 
ing to  Richmann's  rule,  the  amount  of  heat  which  had  appar- 
ently disappeared  ^rom  the  mixture  could  be  calculated.  It 
was  evidently  used  to  melt  the  ice.  The  loss  of  heat  in  this 
case  is  especially  apparent  when  the  experiment  is  tried  in  a 
way  indicated  by  Black.  Suppose  equal  masses  of  ice  and 
water  are  taken,  and  the  temperature  of  the  water  raised  to 
80°  Centigrade.  When  the  ice  is  immersed  jin  the  water,  it 
begins  to  melt,  and  the  temperature  of  the  water  falls  more 
and  more  as  the  melting  proceeds,  until  just  as  the  last  trace 
of  ice  disappears,  the  temperature  of  the  mixtnre  falls  to  0° 
Centigrade.  The  warm  water  in  this  case  has  given  up  80 
calories,  and  this  heat  has  occasioned  no  rise  of  temperature 
in  another  body.  We  conclude  that  it  has  been  used  in  melt- 
ing the  ice. 

Black  considered  that  when  heat  enters  a  body  in  such  a 
way  that  it  causes  a  rise  of  temperature,  it  is  in  such  a  condi- 
tion with  relation  to  the  body  that  it  can  be  detected  by  the 
temperature  sense.  He  therefore  called  it  sensible  heat.  On 
the  other  hand,  the  heat  which  has  passed  into  ice,  and  melted 
it,  cannot  be  detected  by  any  change  of  temperature  which  it 
causes,  and  is  in  a  sense  concealed  in  the  water.  He  therefore 
called  it  latent  heat.  This  term  is  a  very  convenient  one  and 
is  often  used,  although  our  present  conception  of  heat  as  a 
form  of  energy  makes  it  somewhat  inappropriate.  We  may, 
instead  of  it,  use  the  term,  heat  of  fusion. 

The  absorption  of  heat  by  melting  ice  is  an  example  of  what 
occurs  whenever  a  solid  body  melts.  In  every  case  of  melting, 
an  amount  of  heat  is  absorbed,  which  depends  upon  the  mass 
of  the  body  melted  and  upon  the  substance  of  which  it  is  com- 
posed. The  amount- of  heat,  in  calories,  which  is  absorbed  by  a 
kilogramme  of  a  substance,  when  it  melts,  is  called  the  latent 
heat,  or  heat  of  fusion,  of  that  substance.  The  heat  of  fusion 
of  ice  is  about  80  calories. 


HKAT.  153 

Black  perceived  that  when  a  body  melts,  even  though  its 
melting  is  not  brought  about  by  the  entrance  of  heat  from 
without,  it  will  of  necessity  absorb  heat,  and  thus,  if  it  does 
not  receive  heat  from  without,  its  own  temperature  will  fall. 
He  explained  in  this  way  the  behavior  of  freezing  mixtures, 
and  showed  that  the  general  fact  observed  by  Boyle,  that  all 
such  mixtures  melt,  is  the  one  upon  which  their  efficiency  as 
freezing  mixtures  depends. 

Black  also  perceived  that  when  a  liquid  freezes,  it  will  give 
out  an  amount  of  heat  equal  to  that  which  was  absorbed  by  it 
when  it  was  formed  by  melting.  He  explained  in  this  way 
the  constant  temperature  of  a  liquid  while  it  is  freezing,  and 
also  the  rise  of  temperature  which  occurs  when  a  supercooled 
liquid  begins  to  freeze. 

Black  also  .studied  the  case  of  boiling  liquids.  From  the 
constancy  of  their  boiling  points  he  inferred  that  heat  is  ab- 
sorbed by  them  during  the  process  of  boiling.  The  amount  of 
heat,  in  calories,  required  to  turn  a  kilogramme  of  a  liquid 
into  vapor  at  the  boiling  temperature  is  called  the  latent  heat 
of  the  vapor,  or  the  heat  of  vaporization  of  the  liquid.  When 
the  vapor  condenses  again,  it  may  be  made  to  heat  a  quantity 
of  liquid  of  the  same  sort  as  that  from  which  it  is  formed,  and 
thus  to  give  evidence  that  heat  is  emitted  by  a  vapor  on  con- 
densation. By  taking  advantage  of  the  equality  between  the 
heat  of  vaporization  and  the  heat  emitted  on  condensation,  the 
heat  of  vaporization  may  be  determined. 

107.  Calorimeters. — A  calorimeter  is  an  instrument  by 
means  of  which  a  quantity  of  heat  may  be  measured. 

From  the  principles  developed  in  the  previous  sections,  it 
is  evident  that  we  can  measure  the  amount  of  heat  which 
leaves  a  body  by  the  effect  which  it  will  produce  in  some  other 
body.  It  is  not  possible  for  us  to  measure  all  the  heat  which 
a  body  contains. 

The  calorimeter  used  for  the  method  of  mixtures  consists 
of  a  vessel,  isolated  from  surrounding  bodies,  so  far  as  pos- 
sible, so  that  the  heat  which  is  introduced  into  it  will  remain 
in  it  without  change.  A  known  quantity  of  water  is  placed  in 


154  HEAT. 

this  vessel,  and  its  temperature  is  taken.  If  another  body,  say 
a  block  of  iron,  of  known  mass,  is  raised  to  a  known  tempera- 
ture, higher  than  that  of  the  water,  and  if  it  is  then  trans- 
ferred to  the  water,  its  temperature  will  fall  and  the  tempera- 
ture of  the  water  will  rise,  until  they  have  reached  a  common 
value.  This  value  is  then  determined.  The  amount  of  heat 
lost  by  the  iron  in  falling  from  its  original  temperature  to 
the  common  temperature,  is  equal  to  the  amount  of  heat  gained 
by  the  water  in  rising  from  its  original  temperature  to  the 
common  temperature.  The  heat  gained  by  the  water  is 
measured  in  calories  by  the  product  of  the  mass  of  the  water 
and  its  change  of  temperature.  Thus  the  heat  lost  by  the  iron 
is  determined. 

A  calorimeter  employed  by  Black  and  by  Wilcke  consists 
simply  of  a  block  of  ice,  in  which  a  small  cavity  is  made. 
When  the  ice  is  at  zero  temperature,  the  interior  of  the  cavity 
is  dry.  To  keep  it  so,  it  is  covered  with  a  slab  of  ice.  The 
body  whose  heat  capacity  is  to  be  tested  is  heated  to  a  known 
temperature  and  transferred  to  the  cavity.  The  heat  which  it 
gives  up  will  melt  the  ice  around  it.  After  its  temperature 
has  fallen  to  that  of  the  ice,  and  the  ice  no  longer  melts,  the 
water  which  has  been  formed  is  taken  out  and  weighed.  We 
may  state  the  result  obtained  in  terms  of  an  arbitrary  unit 
of  heat,  namely,  the  amount  of  heat  required  to  melt  a  kilo- 
gramme of  ice.  If  this  unit  is  used,  the  weight  of  the  water 
obtained,  in  kilogrammes,  measures  the  amount  of  heat  which 
the  body  has  lost.  As  we  know  that  80  calories  are  required 
to  melt  a  kilogramme  of  ice,  it  is  easy  to  state  this  amount 
of  heat  in  calories. 

This  method  of  melting  has  been  applied  in  several  different 
ways.  The  most  ingenious  of  these  was  that  invented  by 
Bunsen,  who  utilized  th£  change  of  volume,  which  occurs  when 
a  quantity  of  ice.  melts,  as  a  measure  of  the  amount  of  ice 
which  was  melted. 

The  method  of  condensation,  which  has  been  highly  devel- 
oped by  Joly,  measures  the  amount  of  heat,  which  will  pro- 


HEAT.  155 

duce  a  given  change  of  temperature  in  a  body,  by  weighing 
the  amount  of  steam  which  is  condensed  upon  that  body. 

The  method  of  cooling  measures  the  amount  of  heat  which 
leaves  a  body  by  the  rate  at  which  its  temperature  falls.  In 
order  to  carry  out  the  experiment  on  different  substances 
under  similar  conditions,  an  instrument  is  constructed  con- 
sisting of  a  small  polished  box,  in  the  middle  of  which  stands 
the  bulb  of  a  thermometer.  The  box  is  first  filled  with  a  stan- 
dard substance.  It  is  then  raised  to  a  high  temperature  and 
placed  within  a  larger  box,  from  which  the  air  can  be  ex- 
hausted, and  whose  walls  are  kept  at  a  constant  temperature 
by  immersion  in  ice.  The  thermometer  is  observed  from 
minute  to  minute,  and  the  rate  at  which  the  temperature 
changes  for  the  standard  substance  is  thus  determined.  Other 
substances  are  compared  with  the  standard  by  carrying  out 
similar  observations  with  them. 

The  method  of  comparison  depends  upon  the  introduction 
of  equal  quantities  of  heat,  in  any  way  in  which  that  may  be 
done,  into  known  masses  of  a  standard  substance  and  of  the 
substance  under  examination,  and  the  observation  of  their 
changes  of  temperature. 

108.  Specific  Heats. — Calorimetric  observations  are  usually 
employed  to  determine  either  the  heat  capacity  of  a  body,  or 
the  specific  heat  of  a  substance.  The  heat  capacity  of  a  body 
is  supposed  to  be  constant  within  the  range  of  temperature 
employed  in  the  experiment,  and  is  therefore  determined  from 
the  quantity  of  heat  which  leaves  the  body,  when  its  tempera- 
ture undergoes  a  known  change,  by  dividing  that  quantity  of 
heat  by  the  change  in  temperature.  .  The  specific  heat  of  the 
substance  of  which  the  body  is  composed,  provided  it  is  homo- 
geneous, is  obtained  by  dividing  the  body's  heat  capacity  by 
its  mass. 

By  the  study  of  the  specific  heats  of  solid  substances 
through  different  ranges  of  temperature,  it  is  found  that,  as 
a  first  approximation,  they  are  constant  at  all  ordinary  tem- 
peratures. As  a  rule,  the  specific  heat  of  a  substance  increases 
slightly  as  the  temperature  rises.  There  are  a  few  substances, 


156  HKAT. 

of  which  carbon,  in  the  form  of  the  diamond,  is  an  example, 
whose  specific  heat  increases  rapidly  with  rise  of  temperature. 
The  specific  heat  of  the  diamond  is  nearly  three  times  as  great 
at  200°  as  at  0°. 

The  specific  heats  of  liquids  also  vary  in  a  similar  way 
with  the  temperature.  The  most  reliable  observations  indicate 
a  double  variation  in  the  case  of  water,  its  specific  heat  dimin- 
ishing slightly  from  (J°  to  about  35°,  and  increasing  from  that 
point  on. 

The  specific  heat  of  a  substance  in  the  liquid  state  is  always 
greater  than  that  of  the  same  substance  in  the  solid  state. 
The  specific  heat  of  water,  which  has  been  taken  as  standard, 
is  greater  than  that  of  almost  any  other  substance.  So  far 
as  known,  the  only  specific  heats  which  are  greater  than  that 
of  watT  are  thusi-  of  hydrogen,  and  ut  mixtures  of  water  with 
some  <>f  the  alcohol-*. 

The  specific  heats  of  a  gas  differ  considerably  according  to 
the  circumstances  in  which  the  measurement  is  made.  If  the 
gas  is  examined  while  its  volume  is  kept  constant,  its  specific 
heat  will  have  a  certain  Aralue,  called  its  specific  heat  at  con- 
stant volume.  If,  on  the  other  hand,  it  is  examined  while  its 
pressure  is  kept  constant,  so  that  as  its  temperature  rises  it 
expands,  it  is  found  that  an  additional  quantity  of  heat  is 
used  in  raising  it  to  the  same  temperature,  and  its  specific 
heat  is  greater  than  in  the  other  case.  The  specific  heat  thus 
determined  is  called  the  specific  heat  at  constant  pressure. 
Regnault  proved,  by  direct  observation,  that  the  specific  heat 
of  a  gas  at  constant  pressure  is  independent  both  of  the  pres- 
sure and  of  the  temperature  of  the  gas. 

A  very  remarkable  relation  among  the  specific  heats  of 
those-chemical  elements  which  are  found  in  the  solid  state  was 
discovered  by  Dulong  and  Petit.  These  physicists,  examining 
the  specific  heats  of  thirteen  of  the  solid  elements  by  the 
method  of  cooling,  found  that  in  each  case  the  product  of  the 
specific  heat  by  the  atomic  weight  of  the  element  was  approx- 
imately the  same  number.  Now,  the  masses  of  different  ele- 
ments which  contain  the  same  number  of  atoms  are  proper- 


HEAT.  157 

tional  to  their  atomic  weights,  so  that  the  products  of  the 
specific  heats  and  the  atomic  weights  are  the  heat  capacities 
of  masses  containing  the  same  number  of  atoms,  and  since  this 
product  is  the  same  for  many  solid  elements,  we  conclude  that 
for  them  their  atoms  all  have  the  same  capacity  for  heat. 
This  law  was  subsequently  shown  to  apply  approximately  to 
almost  all  the  solid  elements.  The  product  here  defined  is 
called  the  atomic  heat  of  the  element.  Its  value  is  about  6.4. 
The  atomic  heats  of  a  few  of  the  solid  elements,  especially  of 
carbon  and  silicium,  are  exceptions  to  the  general  rule. 

It  was  shown  by  F.  Neumann  and  by  Regnault  that  the 
specific  heats  of  substances  which  are  compounds  of  the  solid 
elements  are  such. as  to  indicate  that  the  atoms  in  composition 
retain  their  atomic  heats.  That  is,  it  is  found  that  the  pro- 
ducts of  the  specific  heats  and  the  molecular  weights  of  com- 
pounds which  have  the  same  number  of  atoms  in  the  molecule 
are  approximately  equal,  so  that  the  molecular  heats  of  such 
compounds  are  equal.  When  the  constituents  of  the  molecules 
are  elements  which  conform  to  Dulong  and  Petit's  law,  the 
quotient  of  the  molecular  heat  divided  by  the  number  of 
atoms  in  the  molecule  is  found  to  be  the  constant  atomic  heat 
already  considered.  When  the  molecule  is  one  which  contains 
atoms  of  an  element  which  cannot  be  examined  in  the  solid 
state,  and  other  atoms  of  elements  which  conform  to  Dulong 
and  Petit's  law,  we  may  calculate  the  atomic  heat  of  the  un- 
known element  from  the  molecular  heat  of  the  compound.  In 
this  way  the  atomic  heat  of  the  gaseous  elements,  when  they 
form  parts  of  the  molecules  of  solids,  have  been  calculated. 
It  is  thus  found  that  the  elements  hydrogen,  nitrogen,  and 
oxygen  do  not  conform,  at  least  in  all  cases,  to  Dulong  and 
Petit's  law. 

109.  Transfer  of  Heat. — It  is  a  matter  of  common  observa- 
tion that  heat  may  be  transferred  from  one  body  to  another. 
Thus,  when  one  end  of  an  iron  bar  is  thrust  in  the  fire,  the 
other  end  gradually  gets  warmer,  and  a  body  may  be  warmed 
by  placing  it  in  front  of  the  fire,  although  no  part  of  it  is  in 
the  fire.  An  experiment  described  by  Newton  proves  that  in 


158  HKAT. 

the  latter  case  the  heat  is  not  transferred  from  the  fire  to  the 
body  by  the  action  of  any  known  material  body  between  them. 
Newton  placed  two  thermometers  in  two  similar  glass  vessels, 
from  one  of  which  the  air  was  exhausted.  After  letting  them 
stand  in  a  cool  place  until  the  thermometers  indicated  the 
same  temperature,  he  transferred  them  to  a  warm  place,  and 
found  that  the  temperature  of  the  thermometer  in  the  vacuum 
rose  nearly  as  fast  as  that  of  the  other  one,  and  that  the  final 
temperatures  of  both  were  the  same.  The  heat  which  reached 
the  thermometer  in  the  vticuum  was  manifestly  translerred  to 
it  from  the  walls  of  the  vessel,  when  they  were  heated  by 
standing  in  the  warm  place,  and  since  there  was  no  known 
material  medium  in  contact  with  the  bulb  of  the  thermometer, 
the  heat  which  it  received  must  have  been  transferred  to  it 
without  the  intervention  of  any  such  medium. 

Heat  transferred  in  this  way  is  said  to  be  transferred  by 
radiation,  and  is  called  radiant  heat.  Its  properties  are  in 
every  respect  like  those  of  light.  Indeed,  subsequent  study 
has  proved  that  radiant  heat  and  light  are  essentially  similar 
in  all  respects,  arid  we  shall  therefore  study  it  in  connection 
with  light.  One  general  principle  was  discovered,  however, 
governing  the  radiation  of  heat,  which  does  not  depend  on  the 
mode  in  which  it  is  propagated,  and  which  may  be  considered 
in  this  place. 

Taking  advantage  of  the  fact  that  radiant  heat  may  be  re- 
flected and  brought  to  a  focus,  as  light  Is,  the  following  experi- 
ment was  tried:  Two  spherical  mirrors  were  set  up  facing 
each  other.  A  thermometer  was  placed  at  the  focus  of  one  of 
them.  At  the  focus  of  the  other  was  placed  a  ball  of  lead 
which  had  been  heated,  though  not  to  redness.  When  this  was 
done,  the  temperature  of  the  thermometer  at  the  other  focus 
began  to  rise,  showing  a  reception  of  heat  by  the  thermometer 
from  the  hot  lead.-  When  a  block  of  ice  was  substituted  for 
the  lead,  the  temperature  of  the  thermometer  fell.  Consider- 
ing these  experiments,  Prevost  perceived  that  the  rational  way 
to  explain  the  fall  of  the  thermometer  produced  by  the  ice, 
was  to  ascribe  it,  not  to  a  radiation  of  cold  from  the  ice,  but 


HEAT.  159 

to  a  radiation  of  heat  from  the  thermometer.  Generalizing 
this  idea,  he  laid  down  the  principle  that  all  bodies  are  at  all 
times  radiating  heat,  and  receiving  heat  from  neighboring 
bodies,  and  that;  the  change  of  temperature  of  the  body  de- 
pends upon  the  relative  amounts  of  heat  which  it  is  receiving 
and  emitting.  When  it  receives  more  than  it  emits,  its  tem- 
perature rises;  when  it  emits  more  than  it  receives,  its  tem- 
perature falls;  when  it  emits  the  same  amount  as  it  receives, 
its  temperature  is  constant.  This  principle  is  known  as  Pre- 
vost's  law  of  exchanges. 

Heat  which  is  transferred  through  a  solid,  when  one  part 
of  it  is  heated,  is  said  to  be  transferred  by  conduction.  The 
experiments  of  Richmann  and  of  Ingenhouss  showed  that  the 
rate  at  which  heat  is  transferred  by  conduction  is  different  in 
different  substances.  It  will  manifestly  depend  also  on  the 
differences  of  temperature  in  the  body,  or  on  the  way  in  which 
the  temperature  of  the  body  changes-  along  the  lines  along 
which  conduction  takes  place.  By  assuming  that  the  flow  of 
heat  along  a  line  is  proportional  to  the  rate  of  change  of  tem- 
perature along  that  line,  Biot,  Rumford,  and  subsequently 
Fourier,  were  able  to  explain  the  movement  of  heat  in  bodies 
in  a  way  which  is  consistent  with  the  results  of  observation. 

In  accordance  with  the  foregoing  assumption,  we  may  de- 
fine the  conductivity  of  a  substance  as  the  amount  of  heat 
which,  in  a  unit  of  time,  will  pass  between  two  unit  areas  in 
the  substance,  so  placed  that  they  stand  at  unit  distance 
apart  and  that  between  them  the  temperature  differs  by  one 
degree. 

When  heat  passes  across  a  surface  at  which  two  substances 
meet,  its  rate  of  transfer,  or  its  surface  conductivity,  depends 
on  the  nature  of  the  substances.  As  a  first  approximation,  it 
is  assumed  to  be  proportional  to  the  difference  of  temperature 
between  the  two  substances.  We  may  define  the  surface  con- 
ductivity as  the  amount  of  heat  which  will  pass,  in  one  unit 
of  time,  through  unit  area  of  the  surface,  when  the  difference 
of  the  temperatures  on  the  two  sides  of  the  surface  is  one 
degree. 


160  HEAT. 

In  most  cases  in  which  bodies  transmit  heat  by  conduction, 
the  temperatures  of  the  different  parts  of  the  body  will  grad- 
ually approach  definite  values.  After  these  definite  values  are 
attained,  no  more  temperature  changes  occur.  In  all  such 
cases  it  is  plain  that  each  part  of  the  body  receives  as  much 
heat  from  the  hotter  portions  of  the  body  as  it  sends  on  to 
the  cooler  portions.  This  condition  of  the  body  is  called  its 
steady  state.  In  many  other  cases,  in  which  the  source  of 
heat  is  not  applied  to  the  body  continuously,  the  temperatures 
of  its  different  parts  vary  continually  in  a  way  which  depends 
on  the  way  in  which  the  heat  is  applied,  on  the  shape  and  size 
of  the  body,  and  on  its  conductivities.  Many  such  cases  have 
been  studied  by  the  help  of  the  assumptions  already  describe;!, 
and  the  theoretical  results  obtained  have  been  found  to  agree 
with  the  results  of  observation. 

The  transfer  of  heat  in  a  liquid  takes  place  generally  by  a 
process  which  is  known  as  convection.  It  is  a  well-known  fact 
that  liquids  receive  heat  readily.  This  was  ascribed  at  first 
to  liquids  being  very  good  conductors.  Rumford  noticed,  how- 
ever, that  masses  of  liquid  suspended  in  fibrous  bodies  take 
up  heat  very  slowly,  and  retain  it  for  a  long  time.  This  ob- 
servation seemed  to  him  inconsistent  with  the  hypothesis  that 
liquids  are  good  conductors,  and  he  accordingly  undertook  an 
investigation  of  the  behavior  of  liquids  when  heated.  Taking 
a  glass  flask  with  a  long  neck,  he  filled  it  with  water,  in  which 
were  suspended  small  particles  or  motes,  and  set  it  in  a  room 
whose  temperature  was  low  and  constant.  When  it  had  stood 
there  until  its  temperature  had  become  that  of  the  room,  and 
until  the  particles  suspended  in  the  water  were  still,  showing 
that  there  were  no  currents  in  the  water,  he  transferred  it  to 
a  warm  room.  The  suspended  particles  at  once  began  to  move 
upward  along  the  walls  and  to  descend  in  the  middle  of  the" 
flask,  showing  that  a  regular  circulation  of  the  water  was 
occurring.  This  circulation  kept  up  until  the  temperature  of 
the  water  had  risen  to  that  of  the  room.  When  he  transferred 
the  flask  to  the  cold  room  again,  currents  in  the  opposite  sense 
occurred,  the  water  moving  downward  along  the  walls  and 


HKAT.  161 

upward  in  the  middle.  After  these  currents  were  once  per- 
ceived, it  was  easy  to  explain  them.  When  the  flask  was 
brought  into  the  warm  room,  its  walls  took  up  heat  from  the 
surrounding  air,  and  heated  the  layers  of  water  which  were 
near  them.  The  density  of  the  water  was  diminished  by  its 
expansion,  due  to  this  heating,  and  it  therefore  rose  along  the 
walls.  The  denser,  because  cooler,  portions  of  the  water  which 
sank  to  give  room  at  the  top  for  the  warmer  portions,  came 
in  turn  in  contact  with  the  walls,  were  also  heated,  and  rose 
along  the  walls.  In  this  way  are  explained  the  rapid  reception 
of  heat  by  a  liquid,  and  the  fact  that  the  temperature  of  a 
liquid  which  is  being  heated  from  the  bottom  is  almost  the 
same  throughout. 

To  test  whether  a  liquid  is  a  good  conductor  in  the  ordi- 
nary sense,  Rumford  tried  to  heat  a  mass  of  water  by  apply- 
ing the  heat  at  the  top,  so  that  the  usual  convection  currents 
could  not  arise.  He  found  so  little  heat  transmitted  to  the 
bottom  of  the  vessel,  that  he  concluded  that  water  did  not 
conduct  at  all.  This  conclusion  is  manifestly  erroneous,  for 
if  there  were  no  conduction  in  water,  the  heating  of  the  water 
near  the  walls  of  the  flask  in  Rumford's  first  experiment,  could 
not  be  accounted  for.  As  was  shown  by  subsequent  observa- 
tion, water  and  other  liquids  are  not  non-conductors  of  heat, 
but  very  poor  conductors. 

Gases  are  ordinarily  heated  by  convection.  It  was  shown 
by  Magnus  that  gases,  like  liquids,  are  poor  conductors. 

110.  Expansion  of  Solids. — The  general  truth  that  a  body 
expands  when  its  temperature  rises  was  illustrated  in  the 
early  work  on  the  subject  of  heat,  by  the  expansion  of  the  air 
in  Galileo's  thermometer,  and  by  the  behavior  of  thermometric 
substances  in  general.  The  Accademia  del  Cimento  demon- 
strated that  metals  expand  when  they  are  heated.  After  the 
discovery  of  the  general  truth,  it  became  a  matter  of  interest 
to  investigate  the  laws  of  this  expansion.  'An  5nstrument 
called  the  pyrometer  was  invented  for  that  purpose,  which 
consisted  essentially  of  a  frame  work  so  arranged  that  one 
end  of  the  bar  of  metal  under  investigation  could  be  kept 


162  11  BAT. 

fixed,  while  the  other  end  was  attached  to  a  rack  and  pinion, 
by  which  a  pointer  could  be  moved  over  a  dial.  The  rod  was 
immersed  in  a  vessel  full  of  water  or  oil,  which  was  heated 
from  beneath.  If  the  temperature  of  the  water  was  raised  and 
the  bar  expanded,  the  pointer  moved  around  the  dial,  and  thus 
measured  the  elongation  of  the  bar.  Inaccurate  as  this  ar- 
rangement was,  it  served  to  show  that  the  elongation  of  a  bar 
of  given  length  is  proportional  to  the  rise  of  temperature 
which  occasions  it,  and  that  bars  of  different  metals  have  dif- 
ferent elongations  for  the  same  rise  of  temperature.  This  lat- 
ter statement  was  illustrated  by  an  experiment  made  by 
DeLuc,  who  clamped  two  bars,  of  iron  and  brass,  lirmly  to- 
gether at  one  end,  and  observed  the  relative  expansion,  when 
the  bars  were  heated,  by  measuring  the  way  in  which  the 
length  of  one  bar  increased  more  than  the  length  of  the  other. 
The  first  accurate  observations  of  the  absolute  elongation 
of  a  bar  were  made  by  Laplace  and  Lavoisier.  In  their  experi- 
ments, one  end  of  the  bar  rested  against  a  massive  stone  pier. 
The  other  end  engaged  with  the  short  arm  of  a  lever  or  sys- 
tem of  levers,  on  the  last  long  arm  of  which  a  mirror  was 
mounted.  While  the  bar  was  at  the  temperature  of  melting 
ice,  the  image  of  a  vertical  scale,  reflected  in  this  mirror,  was 
observed  by  a  telescope.  The  bar  was  then  raised  to  the  tem- 
perature of  boiling  water.  The  consequent  elongation  moved 
the  levers,  and  so  changed  the  position  of  the  mirror.  By 
another  observation  of  the  scale,  and  from  a  knowledge  of  the 
ratios  of  the  arms  of  the  levers,  it  was  possible  to  calculate 
the  elongation  of  the  bar.  It  was  found  that  the  elongation 
of  unit  length  of  the  bar,  produced  by  different  changes  of 
temperature,  was  not  proportional  to  the  change  of  tempera- 
ture in  each  case.  To  represent  the  length  of  fhe  bar  at  dif- 
ferent temperatures  an  expression  involving  the  second,  or 
even  the  third,  power  of  the  temperature  had  to  be  employed. 
It  is  however  true,  as  a  first  approximation,  that  the  elonga- 
tion is  proportional  to  the  rise  of  temperature.  If  we  confine 
ourselves  to  this  approximation,  we  may  define  the  coefficient 


of  expansion  of  a  solid  as  the  elongation  of  unit  length  of  it 
when  it?"  temperature  rises  Irom  0°  to  1°  Centigrade. 

For  most  practical  purposes  this  approximation  is  all  that 
is  needed.  There  is,  however,  one  important  operation, 
namely,  the  construction  and  comparison  of  standards  of 
length,  which  requires  an  accurate  knowledge  of  the  co- 
efficients of  expansion  of  the  bars  of  metal  on  which  those 
standards  are  marked  off. 

111.  Expansion  of  Liquids. — When  a  liquid  expands  by 
heat  in  a  thermometer  bulb  or  in  any  similar  vessel,  the  ex- 
pansion which  is  observed  by  the  rise  of  the  column  is  the 
relative  expansion  of  the  liquid  and  the  vessel.  That  is,  the 
volume  of  the  vessel  increases  as  well  as  the  volume  of  the 
liquid,  and  the  change  of  volume  indicated  by  the  rise  of  the 
column  is  the  difference  between  these  two  changes  of  volume. 
The  absolute  expansion,  which  it  is  sometimes  important  to 
determine,  is  the  actual  increase  in  volume  of  the  liquid. 
DeLuc  observed  the  relative  expansions  of  various  liquids  and 
glass,  and  compared  their  absolute  expansions  by  observing 
the  rise  of  liquid  columns  in  similar  thermometer  tubes.  In 
the  course  of  his  observations  he  discovered  a  remarkable  fact 
in  the  case  of  water.  As  the  temperature  of  the  water  ther- 
mometer rose  from  the  temperature  of  melting  ice,  the  column 
in  the  tube  at  first  fell,  showing  a  contraction  of  the  water. 
At  5°  CVnti-rrHde.  according  to  hi*  observations,  the  column 
readied  its  lowest  point,  and  from  that  temperature  on  it 
rose,  showing  a  regular  expansion.  This  observation  has  been 
repeatedly  confirmed.  The  temperature  at  which  the  volume 
of  the  water  is  least,  or  at  which  its  density  is  greatest,  is 
really  4°  Centigrade.  This  fact,  together  with  tho  fact  that 
ice  is  less  dense  than  water,  plays  an  important  part  in  the 
economy  of  nature;  for  it  is  on  that  account  that  the  temper- 
ature of  the  water  in  large  ponds  and  lakes  rarely  falls  below 
4°,  except  near  the  top. 

The  most  accurate  study  of  the  expansion  of  liquids  has 
been  made  with  the  hydrostatic  balance,  by  determining  the 
apparent  loss  of  weight  of  a  standard  body,  like  a  hollow  glass 


164  HEAT 

sphere,  when  immersed  in  the  liquid  at  different  temperatures. 
If  the  coefficient  of  expansion  of  the  standard  body  is  known, 
the  coefficient  of  expansion  of  the  liquid  may  be  calculated 
from  such  observations.  The  coefficient  of  expansion  in  this 
case  is  denned  as  the  increase  in  volume  of  unit  volume  of  the 
liquid,  when  its  temperature  rises  fn-m  0°  i<>  1°  (.'entiiiniile. 
The  volume  calculated  by  this  coefficient  for  any  temperature 
will  only  be  approximately  correct.  For  accurate  results,  a 
more  complicated  temperature  function  must  be  used. 

112.  Expansion  of  Gases. — The  first  study  of  the  expansion 
of  gases,  or  of  what  amounts  to  the  same  thing,  their  increase 
in  pressure  with  rise  of  temperature,  was  made  by  Amontons. 
His  instrument  we  may  call  an  air-pressure  thermometer.  It 
was  a  glass  globe,  into  the  bottom  of  which  was  inserted  the 
short  limb  of  a  recurved  tube.  Mercury  wag  introduced  into 
the  globe  until  it  was  about  half  filled,  and  so  that  the  top  of 
the  column  in  the  long  limb  of  the  tube  stood  at  the  same  level 
as  that  of  the  mercury  in  the  globe,  when  the  instrument  was 
at  the  standard  temperature  of  melting  ice.  When  this  in- 
strument was  exposed  to  a  higher  temperature,  the  effect  of 
the  expansion  of  the  enclosed  air  was  to  force  down  the  mer- 
cury in  the  globe,  and  so  to  elevate  the  mercury  column  in  tin- 
tube.  Since  the  area  of  the  mercury  surface  in  the  globe  was 
very  many  times  greater  than  that  of  the  cross  section  of  the 
tube,  the  elevation  of  the  mercury  in  the  tube  was  as  many 
times  greater  than  the  depression  of  the  mercury  in  the  globe. 
The  air  in  the  globe  was  thus  subjected  to  pressure,  propor- 
tional to  the  elevation  of  the  mercury  column,  and  its  volume 
was  maintained  almost  unchanged.  If  it  had  been  kept  ex- 
actly constant  by  a  further  increase  of  pressure,  produced  by 
adding  more  mercury  to  the  column,  the  instrument  would 
have  been  a  perfect  air-pressure  thermometer.  As  it  was,  it 
served  very  well  to  enable  Amontons  to  determine  approxi- 
mately the  relation  between  the  rise  of  temperature  and  the 
consequent  increase  in  pressure.  He  stated  that  when  the 
temperature  rose  from  that  of  melting  ice  to  that  of  boiling 


HEAT.  165 

water,  the  increase  in  pressure  was  one-third  the  pressure  at 
the  lower  temperature. 

The  later  attempts  which  were  made  to  obtain  a  measure 
of  the  expansion  of  gases,  were  for  a  long  time  failures.  It 
was  shown  by  Dalton  and  by  Gay-Lussac  that  these  failures 
could  be  traced  to  the  presence  of  water  vapor,  or  rather  of 
water,  in  the  vessel  containing  the  gas.  By  making  the  in- 
terior of  the  vessel  perfectly  dry,  and  by  drying  the  gas,  Gay- 
Lussac  at  last  made  a  successful  study  of  the  expansion  of 
gases.  He  stated  his  results  as  follows: 

1.  All  gases,  whatever  be  their  density,  and  all  vapors,  ex- 
pand equally  for  the  same  change  of  temperature. 

2.  For  the  permanent  gases  the  increase  from  the  ice  point 

•I  f\r\ 

to  the  boiling  point  is  of  the  original  volume 

'266.66 

The  general  law  embodied  in  these  statements  is  known  as 
Gay-Lussac's  law.  The  original  factor  given  in  Gay-Lussac's 
statement  of  the  law  has  been  slightly  modified  by  subsequent 
observations.  We  now  know  that  the  increase  in  volume  of  a 
gas  when  its  temperature  rises  from  the  melting  point  to  the 

1 00 

boiling  point  is  . of  the  original   volume.      The  coefficient  of 

273 

expansion   of  all    gasf>s     is    therefore  - —  .       This    statement    is 

closely  accurate  for  those  gases  which  can  be  condensed  only 
with  great  difficulty.  Those  which  are  easily  condensed  have 
generally  higher  values  of  the  coefficient. 

In  much  of  our  study  of  gases,  we  consider  a  gas  called  the 
ideal  gas,  which  has  no  precise  counterpart  in  nature.  It  is 
defined  to  be  a  gas  which  obeys  Boyle's  and  Gay-Lussac's  laws 
exactly.  Consider  such  an  ideal  gas  confined  in  a  vessel  of 
volume  v  under  the  pressure  p,  «t  the  temperature  0°.  If  the 
temperature  is  lowered  one  degree,  the  volume  will  diminish 

by  -_  part  of  the  volume   at   zero.       If   the   temperature    falls 
273 

two  degree?,  the  diminution  of  volume  will  be  _±_  parts  of  the 

273 


106  IIKAT. 

volume  at  zero;  and  so  on,  as  the  temperature  fulls  lower  an  I 
lower.  When  the  temperature  falls  to  — 273  degrees, the  volume 
of  the  gas  will  vanish. 

This  consequence  of  Gay-Lussac's  law  is  not  so  easily  con- 
ceived as  an  equivalent  result  obtained  by  supposing  the 
volume  of  the  gas  to  be  kept  constant,  and  the  pressure  to 
change  because  of  the  fall  of  temperature.  As  the  temperature 

falls,  the  pressure  diminishes  by  losing   - —   part   of   the    press- 

'27$ 

Ire  at  0°  for  each  fall  of  one  degree,  so  that  when  the  tem- 
perature has  fallen  273  degrees,  the  gas  will  no  longer  exert 
pressure.  This  behavior  of  an  ideal  gas  gives  a  hint  that  there 
may  be  a  temperature  at  which  an  ideal  physical  body  will  lose 
some,  at  least,  of  its  physical  characteristics.  We  have  shown 
that  the  pressure  of  a  gas  may  be  explained  by  ascribing  it  to 
the  movements  of  the  molecules  which  compose  the  gas.  If 
we  accept  this  theory  as  the  true  explanation  of  the  properties 
of  a  gas,  it  follows  at  once  that,  when  a  gas  exerts  no  pressure, 
its  molecules  are  at  rest.  On  this  theory,  therefore,  the  tem- 
perature — 273  is  the  temperature  at  which  the  molecules  of 
an  ideal  gas  would  cease  to  move. 

We  know  from  Boyle's  law  that  the  pressure  of  a  gas  is 
inversely  proportional  to  its  volume,  and  from  Gay-Lussac's 
law  that  the  pressure  is  directly  proportional  to  the  factor 

4-   at,    in   which    a=  -^  is  the  coefficient  of  expansion  and  t 

'2 1 3 

is  the  temperature  on  the  Centigrade  scale.  We  may  combine 
these  proportions  into  an  equation  by  introducing  a  constant 

or   factor   of  proportion,    so    as    to   have  p  =  c  a         If   we 

introduce  in  this  equation  the  numerical  value  of  the  coefficient 

of  expansion,  we  obtain  from  it  the  equation  pv=  [:  (273-H). 

2 1  »"i 

Now   —e—   is   a   constant,  to  which  we  give  the  symbol  R;   and 
2(3 

the  quantity  in  parenthesis  is  the  temperature  on  the  Centi- 
grade scale  increased  by  273.  If,  therefore,  we  suppose  a 


HEAT  167 

thermometer  so  graduated  that  its  zero  indicates  the  tempera- 
ture of  — 273  Centigrade,  and  that  the  length  of  its  degree  is 
the  same  as  that  of  the  Centigrade  degree,  the  temperature 
indicated  by  0°  Centigrade  will  be  indicated  by  273  on  the 
new  scale,  and  any  other  temperature  indicated  by  *  on  the 
Centigrade  scale,  will  be  indicated  by  273-t-t  on  the  new  scale. 
We  designate  temperature  given  in  the  new  scale  by  T.  Tn 
terms  of  this  new  notation,  we  have  the  equation  pv=RT,  as 
a  statement  of  Boyle's  and  Gay-Lussac's  laws. 

In  the  kinetic  theory  of  gases  we  showed  that  the  pressure 
of  a  gas  whose  volume  is  kept  constant  is  proportional  to  the 
mean  kinetic  energy  of  its  molecules.  From  the  equation  just 
obtained,  it  therefore  follows  that  the  temperature  of  a  gas, 
measured  on  this  new  scale,  is  proportional  to  the  mean  kinetic 
energy  of  its  molecules.  Now  the  least  value  which  kinetic 
energy  may  have  is  zero,  and  it  has  no  negative  values.  This 
relation  therefore  indicates  that  there  can  be  no  temperatures 
lower  than  the  temperature  indicated  by  the  zero  of  the  new 
scale,  and  that  this  zero  is  a  limit  of  falling  temperature.  For 
reasons  which  will  subsequently  be  given,  it  is  called  the  abso- 
lute zero,  and  the  scale,  formed  as  we  have  described,  is  called 
the  absolute  scale  of  temperature.  We  have  no  warrant  at 
present  for  the  use  of  such  terms.  We  shall  accordingly  call 
this  zero,  the  zero  of  the  ideal  gas  thermometer,  and  the  scale, 
the  scale  of  the  ideal  gas  thermometer. 

A  thermometer  which  will  indicate  this  scale  very  nearly 
may  be  made  by  enclosing  a  mass  of  dry  air  in  a  vessel  by 
means  of  a  mercury  column,  so  arranged  that,  as  the  volume 
of  the  air  changes,  the  pressure  on  it  may  be  kept  constant. 
On  this  condition,  changes  in  temperature  will  be  proportional 
to  the  changes  of  volume.  Such  an  arrangement,  however,  is 
not  so  satisfactory  in  its  working  as  one  in  which  the  volume 
of  the  air  is  kept  constant,  as  its  temperature  changes,  by 
suitable  changes  of  pressure.  With  this  instrument  the  tem- 
peratures are  taken  proportional  to  the  pressures,  so  that  if 
the  pressure  on  the  air  is  determined  for  a  known  tempera- 


168  HEAT. 

ture,  like  that  of  melting  ice,  any  other  temperature  may  be 
obtained  by  an  observation  of  the  corresponding  pressure. 

A  property  of  gases  which  is  connected  with  the  expansion 
with  rise  of  temperature  was  discovered  by  Erasmus  Darwin 
and  investigated  by  Dalton.  They  found  that  when  air,  or 
any  other  gas,  which  has  been  under  pressure  in  a  closed  ves- 
sel, is  allowed  to  expand  suddenly  into  the  atmosphere,  it  is 
cooled  by  the  expansion,  and  that  if  it  is  suddenly  compressed 
it  will  be  heated.  The  heating  produced  by  compression  may 
be  shown  with  the  so-called  lire-syringe,  the  invention  of  a 
French  workman. 

Another  and  more  important  general  property  of  gases  was 
investigated  by  Gay-Lussac,  and  more  satisfactorily  by  Joule. 
Joule's  experiment  required  the  use  of  two  stout  metallic 
cylinders,  in  one  of  which  air  was  compressed  under  a  high 
pressure.  From  the  other  the  air  was  exhausted.  The  two 
cylinders  thus  prepared  were  connected  by  a  tube,  fitted  with 
a  stop-cock,  which  could  be  opened  to  permit  the  compressed 
air  to  rush  into  the  empty  cylinder.  They  were  immersed  in 
water  in  a  large  calorimeter,  and  the  temperature  of  the  water 
observed.  The  stop-cock  was  then  opened  and  the  air  was 
allowed  to  distribute  itself  uniformly  in  the  two  cylinders. 
Another  observation  of  the  temperature  of  the  water  showed 
no  change  in  it.  As  the  air  had  passed  into  the  empty  cylin- 
der without  doing  external  work,  its  energy  remained  the 
same  in  the  larger  volume  as  it  had  been  in  the  smaller  one. 
Joule  concluded  that  the  energy  of  a  gas  does  not  depend  on 
its  volume.  Since  the  temperature  of  the  air  did  not  change, 
we  know  from  Boyle's  law  that  the  produ-ct  of  its  pressure 
and  volume  did  not  change.  We  also  know  from  the  kinetic 
theory  of  gases  that  this  product  is  proportional  to  the  mean 
kinetic  energy  of  the  molecules  of  air.  Since  this  mean  kinetic 
energy  remains  constant,  we  infer  that  it  alone  is  the  energy 
of  the  gas.  Now,  the  kinetic  energy  of  the  molecules,  and 
therefore  also  the  energy  of  the  gas,  changes  when  the  tem- 
perature changes.  We  may  therefore  state  the  result  obtained 


by  Joule  in  the  law,  that  the  energy  of  an  ideal  gas  is  a  func- 
tion of  its  temperature,  but  is  independent  of  its  volume. 

Later  experiments  by  Joule  and  Lord  Kelvin  showed  that 
a  slight  fall  of  temperature  usually  occurs  when  a  gas  ex- 
pands without  doing  external  work.  The  law  stated  holds  for 
the  ideal  gas. 

113.  Vapors. — If  water  is  exposed  in  an  open  vessel  to  the 
air  it  will  gradually  disappear.  It  is  said  to  have  evaporated, 
and  the  process  by  which  it  disappears  is  called  evaporation. 
Many  other  liquids  evaporate  as  readily  as  water  does,  or  even 
more  readily.  On  the  other  hand,  there  are  many  liquids,  like 
sulphuric  acid  or  mercury,  which  evaporate  so  slowly  that 
their  evaporation  can  hardly  be  detected.  If  the  temperature 
of  the  water  is  observed  while  it  is  evaporating,  it  will  be 
found  to  be  lower  than  that  of  the  surrounding  air.  If  the 
water  is  placed  in  the  receiver  of  an  air  pump,  and  the  air 
around  it  exhausted,  evaporation  will  go  on  much  more  rapidly 
than  under  atmospheric  pressure,  and  the  difference  of  tem- 
perature between  the  water  and  surrounding  bodies  will  be 
much  greater.  During  the  process  heat  is  entering  the  water 
from  without,  and  the  rate  at  which  it  enters  depends  on  the 
difference  of  temperature  between  the  water  and  surrounding 
bodies.  We  therefore  infer  that  the  process  of  evaporation 
involves  the  absorption  of  heat,  and  that  the  amount  of  water 
evaporated  at  a  given  temperature  is  proportional  to  the 
amoxint  of  heat  which  enters  it.  This  conclusion  has  been 
confirmed  by  many  observations. 

The  result  of  the  evaporation  of  water  is  the  production  of 
another  body,  which  is.  called  the  vapor  of  water,  or  water 
vapor.  In  general  it  seems  to  be  similar  to  a  gas.  We  may 
study  the  properties  of  a  vapor  by  inserting  a  small  quantity 
of  the  liquid,  whose  vapor  we  wish  to  examine,  under  the  open 
end  of  a  filled  barometer  tube.  The  liquid  will  rise  through 
the  mercury  of  the  column,  and  will  evaporate  in  the  vacuum 
above.  Ordinarily,  when  the  space  above  is  not  too  great, 
some  of  the  liquid  will  remain  unevaporated  and  will  float  on 
top  of  the  mercury  column.  The  first  thing  to  be  noticed  is 


that,  as  soon  as  the  vapor  is  formed,  the  mercury  column  is 
depressed.  This  indicates  that  the  vapor  is  exerting  a  pres- 
sure upon  the  top  of  the  column.  The  change  in  height  of  the 
mercury  column  is  a  measure  of  this  pressure.  If  the  bar- 
ometer tube  stands  in  30  deep  a  vessel,  and  is  itself  so  long, 
that  we  may  raise  or  lower  it  so  as  to  make  considerable 
changes  in  the  volume  of  the  space  above  the  column,  we  find 
on  changing  the  volume,  that  the  pressure  indicated  by  the 
height  of  the  column  above  the  mercury  surface  outside  the 
tube  remains  constant.  This  is  not  the  way  in  which  a  gas 
would  behave,  for  its  pressure  changes  when  its  volume 
changes.  If  so  little  liquid  has  been  introduced  into  the  tube 
that  none  of  it  remains  unevaporated,  the  pressure  will  be 
less  than  that  obtained  when  liquid  is  present,  and  it  will  not 
remain  constant  when  the  volume  is  changed.  The  vapor  in 
this  case  will  behave  like  a  gas.  It  is  only  when  liquid  as  well 
as  vapor  are  present  in  the  tube  that  the  constant  and  maxi- 
mum value  of  the  vapor  pressure  is  exhibited. 

If  the  instrument,  arranged  as  described,  is  exposed  to  a 
higher  temperature,  the  top  of  the  mercury  column  will  be 
depressed,  showing  an  increase  in  the  vapor  pressure.  On  the 
other  hand,  a  cooling  of  the  instrument  will  show  a  decrease 
in  the  vapor  pressure.  The  rate  of  increase  or  decrease  is 
different  for  different  liquids,  and  has  not  been  found  to  be 
expressible  in  any  general  or  simple  law. 

A  vapor  in  the  condition  in  which  it  exhibits  its  maximum 
pressure  is  called  a  saturated  vapor.  We  may  restate  the  ex- 
perimental results  already  described  by  saying  that  the  press- 
ure of  a  saturated  vapor  is  a  function  of  its  temperature  only, 
being  independent  of  the  volume  occupied  by  the  vapor.  We 
may  explain  this  law  by  supposing  that  a  saturated  vapor  is 
one  in  which  every  unit  of  volume  contains  as  much  vapor  TS 
can  exist  in  it  at  the  given  temperature  without  condensation. 
If  the  volume  of  the  vapor  is  diminished,  enough  of  the  vapor 
condenses  into  the  liquid  state  to  keep  the  density  of  the 
remaining  vapor  the  same  as  before.  If  the  volume  is  in- 


IIKAT  171 

creased,  enough  of  the  liquid  evaporates  to  saturate  the  larger 
volume. 

When  the  volume  of  the  vapor  is  kept  fixed,  while  its  tem- 
perature is  raised,  the  increase  in  pressure  exhibited  by  it  is 
not  due  simply  to  the  increased  energy  of  its  molecules,  but 
also  to  the  fact  that  more  of  the  liquid  evaporates,  so  that  the 
density  of  the  vapor  is  increased.  On  the  other  hand,  when 
the  temperature  is  lowered,  some  of  the  vapor  is  condensed. 

It  was  discovered  by  Dalton  that  when  vapors  of  two  dif- 
ferent liquids  are  formed  in  the  same  volume,  the  maximum 
pressure  which  they  exert  is  equal  to  the  sum  of  the  maximum 
pressures  exerted  by  the  two  vapors  separately.  The  same 
thing  is  true  for  mixtures  of  several  vapors,  or,  if  the  volume 
is  kept  constant,  for  mixtures  of  gases,  or  of  vapors  with  gases. 
In  all  these  cases,  the  pressure  of  the  mixture  is  equal  to  the 
sum  of  the  pressures  of  its  constituents,  if  they  were  to  occupy 
the  same  volume  separately.  This  lawT  is  known  as  Dalton's 
law. 

The  evaporation  which  has  been  described  .takes  place  at 
the  free  surface  of  the  liquid.  Another  mode  of  evaporation 
occurs,  called  ebullition  or  boiling,  in  which  the  evaporation 
occurs  within  the  body  of  the  liquid.  When  the  temperature 
of  the  liquid  has  reached  a  certain  point,  which  depends  upon 
the  pressure  upon  it,  bubbles  of  vapor  appear  in  it,  arising  at 
some  point  on  the  wall  of  the  containing  vessel.  After  boiling 
has  fairly  begun,  these  bubbles  rise  through  the  liquid,  rapidly 
increasing  in  volume  as  they  do  so,  and  break  at  the  top, 
liberating  the  vapor  which  they  contain.  The  temperature  at 
which  this  process  takes  place  generally  depends,  for  a  given 
liquid,  on  the  pressure  upon  it,  but  it  has  been  shown,  by 
Dufour,  that  if  the  liquid  is  prepared  by  previous  boiling  *o 
that  the  air  ordinarily  dissolved  in  it  is  driven  out,  its  tem- 
perature may  be  raised  considerably  above  its  ordinary  boiling 
point,  without  its  boiling.  When  boiling  logins  in  this  case, 
the  formation  of  vapor  is  exceedingly  rapid.  These  phenom- 
ena are  analogous  to  those  observed  when  a  liquid  is  super- 
cool ed*. 


172  HKAT. 

The  striking  similarity  of  vapors  and  gases  suggests  that 
it  may  be  possible  to  explain  the  properties  of  vapors  by  the 
kinetic  theory.  In  order  to  do  so,  we  extend  the  kinetic 
hypothesis  to  liquids,  so  far  as  to  assume  that  the  molecules 
of  a  liquid  are  in  motion,  and  that  they  are  not  all  moving 
with  the  same  velocity.  Their  average  velocity  will  have  u. 
value  depending  on  the  temperature,  but  the  velocities  of  the 
separate  molecules  may,  in  some  cases,  very  much  exceed  this 
average  velocity.  We  apply  this  hypothesis  to  a  liquid  exposed 
in  a  closed  and  otherwise  empty  vessel.  Among  the  molecules 
of  the  liquid,  which  are  moving  in  various  directions,  there 
will  be  some  near  the  upper  surface  which  are  moving  upward, 
and  some  of  these  will  be  moving  with  velocities  which  are 
high  enough  to  carry  them  beyond  the  attraction  of  the  neigh- 
boring molecules  of  the  liquid.  They  then  enter  the  space  above 
as  molecules  of  vapor,  and  in  this  way  the  vapor  is  formed. 
Now  the  molecules  of  the  vapor  are  also  moving  in  various 
directions,  and  some  of  them  will  come  within  the  range  of 
the  forces  of  the  liquid,  and  will  return  to  it.  The  vapor  will 
attain  its  greatest  density  when  it  is  so  dense  that  the  mole- 
cules which  enter  the  liquid  from  the  vapor  are  equal  in 
number  to  those  which  enter  the  vapor  from  the  liquid.  The 
attainment  of  this  condition  plainly  does  not  depend  on  the 
total  number  of  molecules  in  the  vapor,  but  only  on  the  density 
of  the  vapor  just  above  the  liquid,  and  it  therefore  follows 
that  the  maximum  pressure  of  the  vapor  is  independent  of  its 
volume.  Since  the  velocity  of  the  molecules  increases  as  the 
temperature  rises,  there  will  be  more  molecules  at  the  higher 
temperature  whose  velocities  will  be  sufficient  to '  carry  them 
awky  from  the  liquid,  and  consequently  the  density  of  the 
vapor  above  the  liquid  will  have  to  be  greater,  at  the  higher 
temperature,  in  order  that  as  many  molecules  may  leave  the 
vapor  as  enter  it.  Therefore  the  density  and  the  pressure  of 
a  saturated  vapor  will  increase  on  rise  of  temperature. 

114.  Critical  Temperature. — Caignard  de  la  Tour  tried  the 
experiment  of  heating  ether,  when  it  was  sealed  up  in  a  strong 
glass  tube.  Under  these  conditions,  as  the  temperature*  rose, 


HEAT.  173 

the  vapor  in  the  tube  became  more  dense.  The  surface  of 
separation  between  the  liquid  below  and  the  vapor  above,  called 
generally  the  meniscus,  remained  distinctly  visible  until  a  cer- 
tain temperature  was  reached.  At  that  temperature  it  dis- 
appeared, and  the  contents  of  the  tube  became  apparently 
homogeneous.  Before  the  disappearance  of  the  meniscus,  its 
position  in  the  tube  indicated  the  presence  of  a  considerable 
mass  of  liquid,  and  its  disappearance  did  not  seem  to  be  due 
to  gradual  evaporation,  but  to  the  attainment  of  a  condition 
in  which  the  liquid  did  not  dift'er  in  appearance  from  the  vapor 
above  it.  We  may  anticipate  the  discussion  which  is  to  follow 
so  far  as  to  name  the  temperature  at  which  this  change  occurs, 
the  critical  temperature.  For  higher  temperatures  the  con- 
tents of  the  tube  seemed  homogeneous. 

When  the  tube  was  cooled  again,  and  the  critical  tempera- 
ture was  reached,  a  sudden  condensation,  in  the  form  of  a  fine 
fog  or  rain,  took  place  throughout  the  tube,  and  the  liquid 
reappeared. 

Similar  experiments  were  tried  with  other  liquids,  and  for 
many  of  them  the  existence  of  a  critical  temperature  was  de- 
termined. 

If  a  non-saturated  vapor  is  compressed  into  a  small  volume, 
it  will  at  last  reach  the  condition  of  saturation,  and  condensa- 
tion will  begin.  This  fact  suggested  the  possibility  of  con- 
densing the  gases  into  liquids  by  increasing  the  pressure  upon 
them.  For  some  time  attempts  made  in  this  direction  were  un- 
successful. Although  the  gases  in  some  cases  were  subjected 
to  enormous  pressures,  Natterer,  for  example,  using  pressures 
as  high  as  three  thousand  atmospheres,  they  showed  no  signs 
of  liquefaction.  The  first  successful  experiments  were  made  by 
Faraday.  The  method  which  he  employed  involved  the  use  of 
a  strong  glass  tube,  bent  in  the  middle.  Substances  which,  by 
their  chemical  action,  would  produce  the  gas  to  be  examined, 
were  introduced  in  one  of  the  limbs  of  this  tube,  and  the  tube 
was  then  sealed.  The  gas  being  continually  generated,  the 
pressure  on  it  increased.  Taking  a  hint  from  de  la  Tour's 
observations,  Faraday  surrounded  the  end  of  the  tube  which 


1  74  H  EAT. 

did  not  contain  the  substances  generating  the  gas  with  u 
freezing  mixture,  in  order,  if  possible,  to  lower  the  temperature 
of  that  part  of  the  tube  below  the  critical  temperature.  With 
this  arrangement  he  liquified  chlorine,  carbon  dioxide,  and 
many  other  gases.  A  few  of  the  gases,  among  them  hydrogen, 
nitrogen  and  oxygen,  proved  refractory  to  this  treatment. 
Faraday  recognized  that  the  reason  of  this  was  his  inability 
to  obtain  a  temperature  as  low  as  the  critical  temperatures  of 
these  gases. 

This  notion  of  a  critical  temperature  marking  the  highest 
temperature  at  which  a  gas  or  vapor  can  be  condensed  into  a 
liquid  and  marking  a  limit  above  which  a  gas  or  vapor  cannot 
be  condensed  into  a  liquid  by  any  pressure  whatever,  was 
developed  by  Andrews  in  connection  with  his  experiments  on 
the  behavior,  of  carbon  dioxide  under  pressure. 

The  refractory  gases,  or  at  least  some  of  them,  were  con- 
densed by  Cailletet  and  by  Pictet.  The  methods  used  by  these 
investigators  were  essentially  the  same.  The  gas  to  be  con- 
densed was  forced  under  high  pressure  into  a  small  chamber 
furnished  with  a  stopcock,  opening  to  the  air.  After  it  had 
been  cooled  to  the  lowest  attainable  temperature,  the  stopcock 
was  opened,  and  it  was  allowed  to  rush  out.  The  additional 
cooling,  produced  by  the  sudden  expansion,  brought  the  tem- 
perature down  below  the  critical  value,  and  a  little  of  the 
gas  condensed  as  a  liquid,  under  atmospheric  pressure,  on  the 
walls  of  the  tube. 

By  using  more  powerful  refrigerating  agents,  Olszewski, 
Dewar  and  others  have  succeeded  in  lowering  the  temperature 
below  the  critical  value,  while  the  gas  is  under  pressure,  and 
in  this  way  have  obtained  large  quantities  of  nitrogen,  oxy- 
gen and  other  gases  in  the  liquid  state.  By  allowing  the  liquid 
thus  obtained  to  evaporate  rapidly,  its  temperature  is  still 
further  lowered.  In  this  way  nitrogen  has  been  frozen  at 
—220°  Centigrade.  The  lowest  temperature  obtained  in  these 
experiments  is>  estimated  to  be  about  — 255°,  or  about  20  de- 
grees above  the  absolute  zero. 


By  taking  advantage  of  the  cooling  of  a  gas  produced  by 
its  steady  flow  from  an  orifice  without  doing  external  work, 
and  by  cooling  the  gas  which  is  still  under  pressure  by  means 
of  the  cooled  gas,  carried  back  past  the  pressure  chamber, 
Linde  has  succeeded  in  liquefying  air  in  large  quantities  under 
atmospheric  pressure.  The  same  method  has  also  been  used 
for  the  liquefaction  of  other  gases. 

115.  Relations  of  Meat  and  Mechanical  L'ncryy. — In  what 
we  have  studied  up  to  this  time  we  have  ignored  the  question 
of  the  nature  of,  heat.  In  describing  the  phenomena  and  in 
stating  the  laws  of  the  phenomena,  it  has  not  been  necessary  for 
us  to  know  more  about  heat  than  that  it  is  something  which 
is  associated  with  the  sensation  of  temperature,  and  which 
can  be  measured  by  the  calorimeter.  In  this  respect  our 
course  has  been  exactly  similar  to  that  which  was  followed 
in  the  actual  development  of  the  subject,  for  although  the 
conception  of  heat  which  prevailed  during  the  time  of  that 
development  was  given  greater  precision  of  statement  than 
we  have  found  necessary,  yet  after  all,  it  amounted  to  nothing 
more  than  the  conception  we  have  used.  During  the  century 
in  which  most  of  the  experimental  work  was  done,  heat  was 
commonly  believed  to  be  a  substance,  and  hence  indestructible, 
which  by  entering  bodies  raised  their  temperatures.  It  was 
also  supposed  to  exist  in  separate  and  exceedingly  minute 
particles,  which  repelled  each  other,  anu  attracted  the  particles 
of  bodies.  The  additional  hypothesis  that  it  was  not  subject  to 
the  force  of  gravity  was  rendered  necessary  by  the  fact  that 
the  weight  of  a  body  does  not  depend  on  its  temperature.  The 
substance  thus  conceived  of  was  named  caloric  by  Laplace  and 
Lavoisier.  It  is  easy  to  see  that  by  ascribing  the  right  sort  of 
properties  to  such  a  substance,  most  of  the  phenomena  w.hich 
we  have  described  can  be  explained. 

A  few  facts,  however,  were  known,  of  fundamental  import- 
ance for  the  theory,  which  coiild  not  be  explained  on  this 
hypothesis.  Of  these  the  most  important  and  general  one  is 
the  production  of  heat  by  friction.  Lord  Bacon  called  atten- 
tion to  this,  and  made  it  one  of  the  strongest  points  of  the 


176  HKAT. 

argument   in   which   he  contended  that  heat   was  not  a  sub- 
stance, but  the  motion  of  the  minute  parts  of  bodies. 

In  1798  Count  Rumt'ord  published  an  account  of  experi- 
ments which  he  had  made  on  the  heat  produced  by  friction. 
He  was  at  the  time  engaged  in  superintending  the  making  of 
cannon  in  the  arsenal  at  Munich.  His  attention  was  attracted 
to  the  great  quantity  of  heat  developed  in  the  cannon  when 
it  was  being  bored  out,  and  especially  by  the  high  temperature 
of  the  chips  cut -out  by  the  boring  tool.  The  upholders  of  the 
caloric  theory  of  heat  explained  this  high  temperature  by  sup- 
posing that  the  specific  heat  of  the  metal  was  diminished  by 
the  friction  upon  it,  so  that  the  same  quantity  of  heat  in  it 
would  cause  a  higher  temperature.  Rumford  first  showed 
by  direct  measurement  that  the  specific  heat  of  the  chips  cut 
off  by  the  tool  was  the  same  as  that  of  any  block  of  the  same 
metal.  He  then  mounted  a  cylinder  of  the  metal  in  such  a 
way  that  it  could  be  revolved  around  a  blunted  boring  tool, 
which  pressed  against  the  bottom  of  a  deep  cavity  cut  in  it. 
With  this  arrangement  he  found  that  as  the  block  of  metal 
was  turned,  its  temperature  continually  rose.  The  amount  of 
metal  abraded  by  the  blunt  tool  was  so  trifling  that  the  rise 
of  temperature  could  not  reasonably  be  ascribed  to  any 
change  in  its  specific  heat,  and  Rumford  concluded  that  it  was 
due  to  heat  directly  produced  or  made  by  the  friction.  jJy 
surrounding  the  block  of  metal  with  water,  he  was  able  to 
measure  the  quantity  of  heat  thus  produced.  In  Rumford's 
mind  the  essential  feature  of  the  experiment  was  the  contin- 
ued production  of  heat  in  apparently  unlimited  quantities, 
without  any  change  occurring  in  the  bodies  producing  it,  by 
which  it  could  be  accounted  for.  He  stated  his  conclusion  in 
the  following  words:  "It  is  hardly  necessary  to  say  that  any- 
thing which  any  insulated  body,  or  system  of  bodies,  can  con- 
tinue to  furuish  without  limitation,  cannot  possibly  be  a 
material  substance;  and  it  appears  to  me  to  be  extremely 
difficult,  if  not  quite  impossible,  to  form  any  distinct  idea  of 
anything  capable  of  being  excited  and  communicated  in  the 


HEAT.  177 

manner  the  heat  was  excited  and  communicated  in  these  experi- 
ments, except  it  be  motion." 

Almost  at  the  same  time  Davy  performed  experiments  of 
a  similar  nature.  He  constructed  a  mechanism  by  which  two 
blocks  of  ice  were  rubbed  together  in  a  room  whose  tempera- 
ture was  below  the  melting  point.  He  found  that  by  this  opera- 
tion the  ice  was  melted.  He  also  constructed  an  arrangement  by 
which  a  brass  disk  was  turned  by  clock-work  against  the  fric- 
tion of  another  piece  of  brass.  Small  pieces  of  wax  were 
placed  on  the  disk,  to  show,  by  their  melting,  when  the  tem- 
perature had  risen  to  the  melting  point  of  the  wax.  This 
apparatus  he  placed  on  a  block  of  ice  in  the  receiver  of  an  air 
pump,  and  exhausted  the  air  from  around  it.  He  found  that, 
when  the  disk  was  turned  for  a  while,  the  wax  was  melted, 
and  the  arrangement  of  the  experiment  was  such  that  the  heat 
which  appeared  could  not  have  entered  the  disk  from  without. 
•He  drew  from  these  observations  essentially  the  same  conclu- 
sion as  that  drawn  by  Rumford. 

These  experiments  of  Rumford  and  Davy  were  accepted  by 
some  as  proving  that  heat  could  not  be  a  substance,  and 
therefore  that,  as  Thomas  Young  said,  "it  must  be  a  quality, 
and  this  quality  can  only  be  motion."  In  general,  however, 
they  had  no  effect  in  changing  the  prevailing  theoryfthat  heat 
was  a  substance.  The  holders  of  that  theory  either  ignored 
them  altogether,  or  set  them  aside  with  the  expectation  that 
in  the  future  they  would  be  explained  in  a  way  consistent  with 
their  theory. 

In  1842  Robert  Mayer  published  a  paper  in  which  he  as- 
serted the  possibility  of  the  transformation  of  mechanical 
energy  into  heat,  and  the  reversed  transformation  of  heat  into 
mechanical  energy.  His  argument  was  rather  a  metaphysical 
than  a  physical  one,  and  attracted  no  attention.  He  took, 
however,  one  important  step  by  attempting  the  calculation 
of  the  amount  of  heat  which  is  equivalent  to,  or  which  may 
be  transformed  into,  a  unit  of  mechanical  energy. 

In  the  following  year,  Joule  began  an  investigation  of  the 
various  ways  in  which  heat  may  be  produced  by  the  expendi- 


ture  of  mechanical  energy,  in  order  to  determine  whether  or 
not  the  same  amount  of  heat,  in  whatever  particular  way  it 
may  be  produced,  is  always  produced  by  the  expenditure  of 
the  same  amount  of  mechanical  energy.  The  results  obtained 
by  the  different  methods  which  he  employed  were  not  very 
consistent  with  each  other,  though  they  were  of  the  same 
order  of  magnitude.  They  were  sufficiently  consistent,  how- 
ever, to  convince  Joule  that  heat  may  be  produced  by  the 
expenditure  of  mechanical  energy,  and  that  the  amount  of 
heat  produced  is  always  in  the  same  proportion  to  the  amount 
of  energy  expended. 

The  ratio  of  the  energy  expended  to  the  amount  of  heat 
produced  by  it,  or  the  amount  of  energy  which  will  produce 
one  unit  of  heat,  is  called  the  mechanical  equivalent  of  heat. 
Joule  undertook  the  task  of  determining  this  quantity.  To 
do  this  he  used  a  vessel  filled  with  water,  in  which  a  paddle- 
wheel  could  be  revolved.  Flanges  projecting  from  the  walls 
of  the  vessel,  between  which  the  blades  of  the  paddle  passed, 
broke  up  the  circulation  of  the  water,  and  greatly  increased 
the  friction  against  which  the  paddle  turned.  The  paddle  was 
kept  in  motion  by  a  falling  Ayeight.  The  mechanical  energy 
expended,  or  the  work  done  by  the  falling  weight,  was  deter- 
mined from  the  known  value  of  the  weight  and  of  the  dis- 
tance through  which  it  moved,  and  the  heat  developed  by  the 
friction  of  the  paddle  in  the  water  was  determined  by  an  ob- 
servation* of  the  rise  of  temperature.  It  was  found  that  what- 
ever work  was  done,  the  ratio  of  the  work  done  to  the  heat 
developed  was  constant.  Joule  used  also  a  similar  apparatus, 
in  which  the  vessel  and  paddle  used  were  made  of  iron,  and  the 
liquid  used  was  mercury.  He  also  used  an  apparatus  in  which 
the  friction  which  developed  heat  was  that  between  two  iron 
plates  rubbed  together  under  mercury.  From  these  three 
forms  of  the  experiment  he  obtained  the  same  result,  namely, 
that  the  amount  of  work  which  will  raise  a  pound  of  water 
one  degree  Fahrenheit  is  772  foot  pounds.  This  statement  is 
equivalent  to  tTie  following:  That  the  amount  of  work  which 
will  produce  a  calorie  of  heat  is  423  kilogramme-metres.  More 


recent  determinations  have  increased  these  numbers  some- 
what. For  ordinary  work  we  may  take  425  kilogramme- 
metres  as  the  mechanical  equivalent  of  one  calorie. 

Joule's  results  carried  conviction  that  heat  is  a  form  of 
energy,  and  that  it  can  be  transformed  into  mechanical  energy, 
or  mechanical  energy  transformed  into  it,  without  loss. 

110.  Conservation  of  Energy. — Heat  may  be  transformed 
into  work,  or  work  into  heat,  by  processes  which  are  far  more 
complicated  than  the  one  which  we  have  described.  In  certain 
stages  of  these  processes  the  condition  of  the  system  is  often 
such  that  we  do  not  recognize  in  it  the  presence  either  of 
mechanical  energy  or  of  heat.  Whatever  this  condition  may 
be,  it  is  found  that  a  quantitative  relation  exists  between  it 
and  the  work  used,  or  the  heat  absorbed,  to  bring  it  about. 
The  first  of  Joule's  experiments  is  a  good  example  of  one  of 
these  processes.  Joule  began  by  investigating  the  relations  of 
heat  to  the  electric  current,  and  to  the  chemical  action  which 
takes  place  in  the  voltaic  cell,  while  the  current  is  flowing. 
The  chemical  effect  in  the  cells  which  he  used  was  the  con- 
sumption of  zinc  by  the  acid  of  the  cell.  He  first  determined 
the  amount  of  heat  developed  during  the  consumption  of  a 
certain  mass  of  zinc  by  direct  chemical  action,  and  compared 
this  with  the  amount  of  heat  developed  in  an  electric  circuit, 
when  the  same  amount  of  zinc  was  consumed.  He  found  that 
the  two  quantities  of  heat  were  equal.  The  difference  between 
them  was  simply  one  of  distribution,  the  heat  in  the  first  case 
being  developed  directly  at  the  place  where  the  chemical  action 
was  going  on,  while  in  the  second  case  it  was  distributed 
throughout  the  circuit.  Joule  next  inserted  in  the  circuit  a 
small  magnetic  motor.  When  the  motor  was  at  rest,  or  when 
it  was  revolving  without  doing  any  work,  the  heat  developed 
in  the  circuit,  for  the  consumption  of  an  equal  mass  of  zinc, 
was  the  same  as  before.  When,  however,  the  motor  was  made 
to  lift  a  weight  and  so  do  work,  less  heat  was  developed  in  the 
circuit,  for  the  consumption  of  the  same  amount  of  zinc.  The 
amount  of  heat  which  apparently  disappeared  had  evidently 
been  transformed  into  mechanical  work.  The  process  by  which 


180  HBAT. 

this  transformation  had  taken  place  was  not,  however,  so 
direct  a  one  as  that  in  which  heat  is  produced  by  friction.  It 
involved  a  series  of  intermediate  conditions  of  the  system, 
which  were  neither  mechanical  nor  thermal  conditions.  To 
get  a  full  survey  of  this  operation,  we  should  go  back  to  the 
fact  that  the  zinc  used  in  the  cell  was  separated  from  an  inert 
chemical  combination  by  the  aid  of  heat.  Beginning  at  that 
point,  the  operation  described  was  one  in  which  the  heat  used 
to  obtain  the  pure  zinc  was  transformed  into  mechanical 
work. 

To  explain  this  series  of  conditions,  and  countless  others 
like  them,  we  assume  that  energy  exists  in  the  universe  in 
several  forms,  and  that  the  energy  in  any  one  form  can  be 
transformed  into  any  other.  Thus,  in  the  example  before  us, 
the  heat  originally  used  in  the  furnace  was  turned  into  energy 
of  possible  chemical  combination,  resident  in  the  zinc  and  the 
acid;  this  energy  was  transformed  into  electrical  energy,  and 
this,  in  turn,  either  into  heat  or  into  mechanical  work.  So 
far  as  we  can  judge  by  experiment,  transformations  of  enei'gy 
take  place  without  any  loss  or  gain  of  energy. 

The  general  principle  that  there  exists  in  the  universe  a 
certain  quantity  of  energy,  which  is  unchanged  by  any  nat- 
ural operation,  was  first  announced  by  Mayer.  It  was  also 
announced  by  Joule  on  independent  grounds,  and  received  its 
first  confirmation  from  his  experimental  work.  The  publica- 
tion by  Helmholtz,  in  1847,  of  an  important  paper,  in  which 
he  discussed  the  principle  in  connection  with  its  application 
to  various  departments  of  physics,  and  showed  that  it  was 
consistent  with  all  that  was  known  about  natural  operations, 
may  be  considered  to  mark  the  establishment  of  the  principle. 

In  its  general  form,  the  principle  is  known  as  the  conserva- 
tion of  energy.  It  may  be  stated  as  follows:  In  any  closed 
system,  that  is,  in  any  system  into  which  no  energy  enters  and 
out  of  which  no  energy  goes,  the  amount  of  energy  remains 
constant,  whatever  be  the  transformations  going  on  within 
the  system.  To  make  this  statement  accord  more  nearly  with 
the  conditions  under  which  experiments  must  be  carried  on, 


HEAT.  181 

we  may  say  that  the  amount  of  energy  in  any  system,  what- 
ever be  the  transformations  which  go  on  within  the  system,  is 
increased  or  diminished  only  by  an  amount  equal  to  that  of 
the  energy  which  passes  through  the  bounding  surface  of  the 
system.  A  statement  which  we  cannot  contradict,  although 
it  is  one  which  we  cannot  prove,  is  that  the  energy  of  the 
universe  is  a  constant  quantity. 

117.  Kinetic  Theory  of  Heat. — As  an  example  of  the  use 
of  the  inductive  method  of  reasoning,  Lor,d  Bacon,  in  the 
Novum  Organum,  considered  the  nature  of  heat.  On  grounds 
which  would  now  be  considered  very  unsatisfactory,  he  came 
to  the  conclusion  that  heat  was  a  peculiar  kind  of  motion  of 
the  small  particles  of  bodies.  Xewton,  and  his  contemporary, 
Locke,  held  the  same  view.  The  theory  that  heat  is  an  im- 
ponderable substance  was,  however,  so  easy  to  work  with,  and 
satisfied  so  fully  most  of  the  demands  made  upon  it  by  the 
discoveries  of  the  time,  that  it  displaced  the  other  theory,  ex- 
cept in  the  minds  of  a  few  leading  thinkers,  like  Cavendish  and 
Thomas  Young.  We  have  seen  how  Rumford  and  Davy  were 
led  to  adopt  a  view  similar  to  Bacon's  by  reflecting  upon  the 
results  of  their  experiments.  Joule's  thought  on  the  matter 
took  the  same  course.  When  it  was  proved  by  Clausius  and 
Maxwell  that  the  properties  of  gases  could  be  explained  by 
the  kinetic  theory,  it  became  almost  a  matter  of  course  to  ex- 
tend this  theory  to  all  the  states  of  matter,  and  to  account  for 
their  relations  to  heat  by  the  motions  of  their  molecules. 

The  method  employed  by  Mohr  may  be  used  to  confirm  our 
belief  in  the  general  kinetic  theory  of  heat.  It  consists  in 
assuming  that  the  molecules  of  all  bodies  are  in  motion,  and 
that  this  motion  is  increased  when  heat  enters  the  body,  and 
in  tracing  the  consequences  of  this  assumption  to  see  whether 
it  will  afford  reasonable  explanations  of  the  properties  of 
bodies  with  respect  to  heat.  For  example,  we  may  explain 
the  expansion  of  bodies  by  heat,  by  ascribing  it  to  their  in- 
creased molecular  motions,  by  means  of  which  each  molecule 
is  able  to  force  its  neighbors  away  from  itself  a  little  farther. 
We  explain  conduction  of  heat  as  the  transfer  of  kinetic  energy 


182  UKAT. 

by  impact  from  the  hotter  or  more  rapidly  moving  molecules 
to  the  colder  ones  near  them.  We  explain  evaporation  in  the 
way  which  has  already  been  given  in  §113.  We  explain  the 
cooling  of  a  gas  when  it  expands  against  external  pressure,  by 
supposing  that  the  work  that  is  done  by  that  expansion  is  de- 
rived from  the  kinetic  energy  of  its  molecules.  We  explain 
the  sensation  of  heat  by  supposing  it  due  to  the  impacts  of 
the  molecules  against  the  ends  of  the  nerves.  In  general,  when 
mechanical  work  is  transformed  into  heat,  we  suppose  it  to  be 
transformed  into  the  kinetic  energy  of  the  molecules  of  the 
heated  body,  and  into  the  potential  energy  which  those  mole- 
cules acquire  by  reason  of  the  expansion  of  the  heated  body. 
There  are  some  operations,  as,  for  example,  the  melting  of  a 
solid,  which  cannot  be  explained  completely  without  additional 
assumptions.  These  assumptions,  however,  are  not  incon- 
sistent with  the  kinetic  theory,  but  extend  it  by  considering 
the  motion  of  the  atoms  as  well  as  that  of  the  molecules. 

So  long  as  we  consider  only  heat  and  mechanical  energy, 
we  may  obtain  a  consistent  account  of  their  relations  by  con- 
sidering kinetic  energy  to  be  the  only  real  energy.  For  if  we 
assume  that  there  exists  another  medium  or  form  of  matter, 
which  our  senses  cannot  perceive,  and  that  this  medium  is  in 
motion,  we  may  explain  all  forms  of  the  potential  energy  of 
bodies  by  supposing  them  due  to  the  kinetic  energy  of  a  larger 
system,  consisting  of  those  bodies  and  of  this  hypothetical 
medium.  The  tendency  in  physical  speculation  has  been,  for 
some  time,  to  attempt  to  explain  all  physical  phenomena  by 
ascribing  them  to  the  motions  either  of  tangible  bodies  or  of 
this  hypothetical  intangible  medium.  That  is,  the  attempt  has 
been  made  to  construct  what  may  be  called  a  mechanical  model 
of  the  physical  universe.  Recent  researches,  especially  those 
connected  with  the  electrical  discharge  and  the  behavior  of 
radioactive  bodies,  have  indicated  the  possibility  of  explaining 
the  energy  of  matter  by  ascribing  it  to  the  motions  in  a 
medium  of  separate  portions  of  electricity.  This  explanation 
has  not  yet  been  worked  out,  but  it  is  at  least  a  possible  one, 


HEAT.  183 

and  there  is  little  doubt  that  for  some  time  speculative  physics 
will  work  along  the  new  line  indicated  by  it. 

118.  Laics  of  Thermodynamics. — When  heat  is  transformed 
into  work,  the  transformation  is  effected  by  means  of  a  body 
which,  in  the  discussion  that  follows,  we  shall  call  the  working 
body.  In  order  to  study  the  transformation  with  as  little 
complication  as  possible,  we  consider  it  effected  by  a  series  of 
changes  which  bring  the  physical  condition  of  the  working 
body  and  of  the  other  parts  of  the  system  back  again  to  that 
in  which  they  were  before  the  operation  began.  Such  an 
operation  is  called  a  cyclic  operation,  or  a  cycle.  At  the  end 
of  a  cycle  the  energy  of  the  working  body  is  the  same  as  it 
was  at  the  beginning,  so  that  the  heat  which  the  working  body 
has  received  has  been  either  given  out  by  it,  or  transformed 
into  work.  The  use  of  such  cycles  was  introduced  by  Sadi 
Carnot,  in  1824.  The  simple  form  of  cycle,  also  proposed  by 
Carnot,  we  shall  call  a  Carnot  cycle.  In  performing  it  the 
working  body  is  used  in  connection  with  two  limitless  bodies, 
at  different  temperatures,  with  which  the  working  body  can 
exchange  heat.  The  hotter  of  these  two  bodies  is  called  the 
source,  the  cooler,  the  refrigerator.  This  set  of  bodies  is  called 
a  thermodynamic  engine.  To  put  the  engine  through  a  Carnot 
cycle,  we  take  the  working  body  at  the  temperature  of  the 
refrigerator,  and  compress  it  while  it  is  enclosed  in  an  envelope 
impermeable  to  heat.  Its  volume  will  diminish,  and  its  tem- 
perature will  rise,  while  mechanical  work  is  done  upon  it  from 
without.  This  first  operation  is'  called  an  adiabatic  com- 
pression. Compression  is  continued  until  the  temperature  of 
the  working  body  rises  to  that  of  the  source.  The  working 
body  is  then  put  in  contact  with  the  source,  and  allowed  to 
expand.  As  it  does  so,  heat  flows  in  from  the  source,  so  as  to 
keep  the  temperature  of  the  working  body  constant.  At  the 
same  time  the  working  body  does  work  by  expanding  against 
external  pressure.  This  operation  is  called  an  isothermal  ex- 
pansion. It  is  allowed  to  continue  until  any  desired  quantity 
of  heat  is  withdrawn  from  the  source.  The  working  body  is 
then  enclosed  in  an  envelope  impermeable  to  heat,  and  Is 


184  HEAT. 

allowed  to  expand  still  further.  During  this  adiabatic  ex- 
pansion, it  does  additional  work  against  external  pressure  and 
its  temperature  falls.  The  expansion  is  allowed  to  continue 
until  the  temperature  of  the  working  body  becomes  that  of 
the  refrigerator.  The  working  body  is  then  put  in  contact 
with  the  refrigerator,  and  is  compressed.  Heat  flows  from  it 
into  the  refrigerator,  so  as  to  keep  its  temperature  equal  to 
that  of  the  refrigerator.  Work  is  done  upon  it  during  this 
operation.  This  isothermal  compression  is  continued  until 
the  volume  of  the  working  body  becomes  again  that  which  it 
was  at  the  beginning  of  the  operations.  The  working  body  i^ 
thus  brought  to  the  same  condition  as  that  in  which  it  was 
at  first,  and  the  series  of  operations  which  it  has  performed 
constitutes  a  cycle. 

In  performing  the  second  and  third  operations  of  the  cycle, 
the  working  body  has  done  external  work;  in  performing  the 
fourth  and  first  operations,  it  has  had  work  done  on  it.  The 
whole  amount  of  work  done  during  the  cycle  is  the  difference 
between  these  two  quantities,  and  is  positive ;  that  is,  the  work- 
ing body  has  performed  external  work. 

In  the  study  of  thermodynamics,  the  principle  of  the 
equivalence  of  heat  and  energy  is  called  the  first  law  of  ther- 
modynamics. It  may  be  stated  as  follows:  When  heat  is 
transformed  into  work,  or  work  into  heat,  the  quantities  of 
heat  and  work,  when  measured  in  the  proper  units,  are  equal 
to  each  other.  When  we  apply  this  law  to  the  cyclic  opera- 
tion just  described,  it  is  evident  that  the  work  done  during 
the  cycle  must  have  been  due  to  the  transformation  of  heat 
in  the  working  body,  and  since  the  heat  in  the  working  body 
is  the  same  at  the  end  of  the  cycle  as  it  was  at  the  beginning, 
the  heat  transformed  must  have  been  part  of  that  received 
from  the  source.  The  work  done  is  therefore  equal  to  the 
difference  between  the  heat  which  leaves  the  source  and  the 
heat  which  enters  the  refrigerator. 

The  cyclic  operation  which  has  been  described  is  reversible. 
That  is,  the  series  of  operations  can  be  described  in  the  reverse 
order,  and  so  that  every  feature  of  the  cycle  is  exactly  re- 


HKAT.  185 

versed.  The  fourth  operation,  for  example,  becomes  on  re- 
versal an  isothermal  expansion,  by  which  the  same  amount  of 
heat  is  received  by  the  working  body  at  the  temperature  of  the 
refrigerator  as  was  sent  into  the  refrigerator  during  the 
isothermal  compression,  and  the  work  done  by  the  working 
body  during  this  expansion  is  equal  to  the  work  done  on  it 
during  the  former  compression.  Similarly  it  may  be  shown 
that  the  other  steps  of  the  cycle  are  also  reversible. 

Clausius  and  Lord  Kelvin  proved  a  most  important  theorem 
regarding  the  operation  of  a  reversible  cycle.  Carnot  first 
asserted  it,  but  his  demonstration  of  it  was  erroneous.  It  is 
generally  known  as  Carnot's  theorem.  We  may  state  it  as 
follows :  The  efficiency  of  any  reversible  engine  is  greater  than 
that  of  any  other  engine  working  between  the  same  source  and 
refrigerator.  By  efficiency  is  meant  the  ratio  of  the  work  done 
by  the  engine  to  the  heat  received  from  the  source.  The  proof 
of  this  theorem  depends  upon  a  general  principle,  which  we 
shall  call  the  second  law  of  thermodynamics.  To  lead  up  to 
this  principle,  let  us  consider  a  cyclic  operation  performed  by 
two  engines,  one  of  which  is  reversible,  while  the  other  is  not. 
We  assume,  for  the  sake  of  argument,  that  the  non-reversible 
engine  is  more  efficient  than  the  other  one,  and  we  gear  the 
engines  together  so  that  the  non-reversible  engine,  working 
forward,  spends  all  its  work  in  driving  the  reversible  engine 
backward.  On  the  whole,  then,  no  work  will  be  done  by  the 
two  engines.  We  have  supposed  the  engine  which  works  for- 
ward to  be  the  more  efficient.  It  will  therefore  have  to  take 
from  the  source  less  heat  to  do  a  certain  amount  of  work  than 
the  reversible  engine  will,  and  therefore  when  that  work  is 
spent  in  driving  the  reversible  engine  backward,  the  reversible 
engine  gives  up  to  the  source  more  heat  than  the  non-reversible 
engine  takes  from  it.  Since  the  first  law  applies  to  both 
engines,  it  follows  that  the  non-reversible  engine  delivers  less 
heat  to  the  refrigerator  than  the  reversible  one  takes  from  it 
when  it  is  run  backward.  The  final  result  of  the  performance 
of  such  a  cycle  as  this,  which  can  be  repeated  as  many  times 
as  we  please,  is  to  transfer  heat  from  the  refrigerator  to  the 


186  HEAT. 

source,  while  at  the  same  time  no  work  is  done,  and  no  change 
occurs  in  the  internal  condition  of  the  bodies  which  constitute 
the  system.  The  result  thus  reached  would  require  us  to  be- 
lieve that  heat  would  pass  of  itself,  or  without  any  comper- 
sating  change  in  the  physical  condition  of  the  system,  from 
the  colder  parts  of  the  system  to  its  hotter  parts.  Now,  in 
our  experience,  heat  does  not  behave  in  this  way.  It  passes 
always  from  hotter  bodies  to  colder  bodies,  and  not  in  the 
reverse  sense. 

Clausius  embodied  the  general  belief  on  this  subject  in  the 
following  statement,  called  the  second  law  of  thermodynamics- 
Heat  cannot  pass  of  itself,  or  without  compensation,  from  a 
colder  to  a  hotter  body. 

If  we  accept  this  law  as  valid,  we  deny  the  possibility  of 
the  result  obtained  by  the  use  of  the  two  engines  whose 
operation  we  have  described,  and  we  accordingly  deny  the 
validity  of  the  hypothesis  upon  which  that  operation  depended, 
namely,  the  hypothesis  that  the  efficiency  of  the  non-reversible 
engine  was  greater  than  that  of  the  reversible  engine.  The 
efficiency  of  the  reversible  engine  is  therefore  the  greatest 
possible. 

It  is  easy  to  extend  this  demonstration  so  as  to  show  that 
the  efficiency  of  all  reversible  engines,  which  work  between  the 
same  source  and  refrigerator,  is  the  same.  The  efficiency  of 
the  reversible  engine  is  therefore  independent  of  the  particular 
material  used  in  the  working  body.  It  depends  only  on  the 
temperatures  of  the  source  and  the  refrigerator.  In  other 
words  the  efficiency  of  the  Carnot's  engine  is  a  function  of  the 
temperatures  of  the  source  and  refrigerator. 

119.  Absolute  Scale  of  Temperature. — In  the  definition  of 
the  scales  of  temperature  which  we  have  hitherto  used,  we 
avail  ourselves  of  some  property  of  a  body  which  is  a  function 
of  its  temperature.  For  example,  in  our  definition  of  the 
Centigrade  scale,  we  make  use  of  the  volume  of  mercury  con- 
tained in  glass.  In  this  instance,  as  in  all  the  others,  the  prop- 
erty employed  is  that  of  a  particular  body,  or  at  best,  of  a 
particular  substance. 


HEAT.  1ST 

We  have  now  discovered,  by  the  study  of  the  Carnot  cycle, 
a  property,  depending  only  on  temperature,  which  is  not  pecu- 
liar to  any  one  body  or  to  any  one  substance,  but  is  common 
to  all.  We  may  therefore  use  this  property  as  a  means  of 
denning  a  scale  of  temperature,  in  such  a  way  as  to  make  our 
definition  independent  of  any  particular  substance.  The  scale 
which  we  shall  thus  define  is  called  the  absolute  scale  of  tem- 
perature. 

A  complete  definition  of  the  absolute  scale  is  obtained  by 
the  choice,  first,  of  the  form  of  the  temperature  function 
which  measures  the  efficiency,  and  secondly,  by  a  determina- 
tion of  the  length  of  the  degree,  or  of  the  numerical  value  of 
some  standard  temperature,  on  that  scale. 

One  general  remark  may  here  be  made,  which  holds  true, 
whatever  be  the  form  of  the  temperature  function  chosen. 
The  efficiency  of  an  engine  working  from  any  given  source  will 
increase  as  the  temperature  of  the  refrigerator  falls;  but  it 
can  never  exceed  unity,  for  it  is  manifestly  impossible  for 
more  work  to  be  done  by  the  engine  than  is  equivalent  to  the 
heat  received  by  it  from  the  source.  The  temperature  of  the 
refrigerator  for  which  the  efficiency  becomes  unity  is  there- 
fore the  lowest  conceivable  temperature.  There  can  be  no 
lower  temperature  than  that,  and  it  must  be  the  same  in  all 
bodies  which  will  serve,  when  used  as  refrigerators,  to  make 
the  efficiency  equal  to  unity.  This  temperature  is  therefore  a 
natural  zero,  and  it  is  advisable,  in  constructing  the  absolute 
scale,  to  mark  it  as  the  zero  point  of  that  scale.  All  actual 
temperatures  will  then  be  positive. 

Turning  now  to  the  choice  of  the  particular  temperature 
function  by  which  the  efficiency  shall  be  expressed,  we  are 
guided  in  making  it  by  the  study  of  the  efficiency  of  an  engine 
in  which  the  ideal  gas  is  used  as  the  working  body.  If  we 
express  the  temperatures  of  the  source  and  refrigerator  on 
the  scale  of  the  ideal  gas  thermometer,  we  find  that  the  effi- 
ciency of  such  an  enfine  is  equal  to  the  difference  between 
the  temperatures  of  the  source  and  of  the  refrigerator,  divided 
by  the  temperature  of  the  source.  That  is,  if  we  represent 


188  HEAT. 

these  temperatures  by  8  and  R  respectively,  the  efficiency  is 

expressed   in   terms   of   them  by   the   formula,    S      R.        Since 

S 

the  efficiency  does  not  depend  on  the  nature  of  the  work- 
ing body,  this  expression  is  also  the  measure  of  the  efficiency 
when  any  other  substance  than  the  ideal  gas  is  used  as  the 
working  body.  This  relation  so  commends  itself  by  its  sim- 
plicity, that  we  adopt  it  as  the  general  relation,  or  form  of 
the  temperature  function,  by  which  the  efficiency  is  ex- 
pressed. That  is,  we  suppose  the  efficiency  to  be  known,  and 
define  the  temperatures  of  the  source  and  refrigerator  by  a 
formula  similar  in  form  to  that  just  given. 

From  this  formula  it  appears  that  the  efficiency  of  the 
engine  becomes  unity  when  the  temperature  of  the  refrigerator 
becomes  the  zero  of  the  ideal  gas  thermometer.  This  zero  is 
therefore  the  lowest  conceivable  temperature,  and  we  may 
adopt  it  as  the  absolute  zero,  or  zero  of  the  absolute  scale. 

It  remains  to  adopt  the  length  of  a  degree  on  the  absolute 
scale.  It  is  convenient  to  have  this  length  as  nearly  as  pos- 
sible the  same  as  the  length  of  the  Centigrade  degree.  We 
accordingly  adopt  100  degrees  as  the  difference  of  absolute 
temperature  between  the  melting  point  and  the  boiling  point. 
By  experiment  we  know  that  the  efficiency  of  an  engine  work- 
in  A 

ing   between   these    temperatures    is   _!_   and   hence,    using   the 

373 

general  formula  for  efficiency  and  the  convention  just  made, 
we  obtain  373  for  the  absolute  temperature  of  boiling  water 
and  273  for  the  absolute  temperature  of  melting  ice.  After 
the  numerical  value  of  one  temperature  is  fixed,  the  numerical 
value  of  any  other  temperature  is  found  by  determining  the 
efficiency  of  an  engine  working  between  the  two  temperatures, 
and  using  the  general  formula.  The  expansion  of  mercury  in 
glass  is  so  nearly  proportional  to  the  change  of  absolute  tem- 
perature, that  we  may  obtain  a  close  approximation  to  the 
absolute  temperature  of  a  body  by  adding  273  to  its  tempera- 
ture on  the  Centigrade  scale. 


HEAT.  189 

We  have  thus  obtained  a  complete  scale  of  temperature, 
starting  from  the  absolute  zero,  and  determined  in  a  way 
which  is  independent  of  the  properties  of  any  particular  sub- 
stance. •  It  is  the  same  as  the  scale  of  the  ideal  gas  thermom- 
eter, but  has  not  the  hypothetical  character  of  that  scale. 
The  kinetic  theory  of  gases  indicates  that  absolute  tempera- 
ture is  a  measure  of  the  kinetic  energy  of  the  molecules  of  a 
gas,  and  perhaps  of  other  bodies.  On  this  view  the  absolute 
zero  is  the  temperature  of  a  body  whose  molecules  are  at  rest. 

120.  Thcrmodynamic  Properties  of  Bodies. — When  the 
properties  of  bodies  are  examined  by  the  methods  of  thermo- 
dynamics, many  important  relations  are  found  among  them, 
which  are  confirmed  by  experiment.  For  example,  it  may  be 
proved,  as  a  general  principle,  that  a  substance,  which  ex- 
pands under  constant  pressure  when  its  temperature  rises, 
has  its  temperature  raised  by  an  adiabatic  compression;  and 
that  a  substance,  which  contracts  when  its  temperature  rises, 
has  its  temperature  lowered  by  adiabatic  compression.  This 
general  conclusion  was  confirmed  by  Joule  for  the  case  of 
water.  In  the  intorval  between  <>°  und  4°  Ontiirniclp.  within 
which  water  contracts  as  its  temperature  rises,  a  sudden  com- 
pression will  lower  its  temperature.  At  any  temperature 
higher  than  4°,  a  sudden  compression  will  raise  its  temperature. 

As  another  example,  we  may  take  the  way  in  which  the 
melting  point  of  ice  depends  upon  pressure.  It  may  be  proved, 
as  a  general  principle,  that  the  melting  temperature  of  a  sub- 
stance, that  is,  the  temperature  at  which  the  solid  and  liquid 
states  nf  the  substance  are  in  equilibrium,  will  rise  if  the 
pressure  is  increased,  provided  the  relative  density  of  the 
solid  is  greater  than  that  of  the  liquid;  but  it  will  fall  if  the 
pressure  is  increased,  provided  the  relative  density  of  the  solid 
is  less  than  that  of  the  liquid.  This  principle  was  confirmed 
for  the  case  of  water  by  the  experiments  of  James  Thomson 
and  his  brother,  Lord  Kelvin.  Water  is  a  substance  belonging 
to  the  second  class,  for  which  the  density  of  the  solid  is  less 
than  that  of  the  liquid.  When  a  mixture  of  water  and  ice, 
whose  temperature  under  atmospheric  pressure  is  0°  Centi- 


1!)0  HEAT. 

grade,  was  subjected  to  additional  pressure,  its  temperature 
fell.  The  temperature  rose  again  when  the  pressure  was  re- 
moved. By  using  this  principle  we  explain  the  melting  of  ice 
under  pressure,  and  its  return  to  the  solid  state,  or  its  rege- 
lation,  when  the  pressure  is  removed.  Other  substances,  for 
example,  paraffine,  which  belong  to  the  first  class,  were  shown 
by  Bunsen  and  by  other  observers  to  exhibit  a  rise  of  temper- 
ature when  pressure  was  applied  to  them. 

The  methods  of  thermodynamics  have  been  applied,  not 
only  to  the  study  of  the  general  properties  of  bodies,  but  more 
particularly  to  the  study  of  solutions,  and  of  mixtures  of 
different  substances  which  act  on  each  other  chemically.  In 
this  way  a  foundation  has  been  laid  for  the  physical  study  of 
chemical  action.  It  should  be  said  that  the  results  which  have 
been  obtained  do  not  furnish  an  explanation  of  chemical 
action.  They  merely  enable  us  to  classify  chemical  actions 
under  certain  general  statements,  which  depend  upon  the 
validity  of  the  second  law  of  thermodynamics.  An  explanation 
of  those  actions  would  require  an  explanation  of  that  law. 

121.  Dissipation  of  Energy. — According  to  tlit-  general 
principle  of  conservation,  the  energy  of  the  universe  is  con- 
stant. Not  all  of  it,  however,  is  available  for  doing  work.  In 
all  cases  of  the  transformation  of  energy  in  which  non-con- 
servative forces  act,  some  at  least  of  the  energy  appears  in 
the  form  of  heat.  If  such  transformations  were  continued 
long  enough,  all  other  energy  in  the  universe  would  be  trans- 
formed into  heat,  and  if  this  heat  could  be  transformed  into 
mechanical  energy,  all  the  energy  of  the  universe  would  be 
available  for  use  as  mechanical  energy.  But  this  is  not  the 
case.  We  know,  from  our  study  of  the  Carnot  engine,  that 
even  when  heat  is  transformed  into  work  in  the  most  efficient 
way  possible,  some  of  the  heat  is  not  transformed,  but  is  trans- 
ferred from  a  higher  to  a  lower  temperature.  None  of  the 
heat  in  the  body  at  lower  temperature  can  be  utilized,  unless 
a  still  colder  body  can  be  found  to  serve  as  refrigerator.  It 
is  plain,  therefore,  that  the  final  effect  of  all  transformation's 
of  energy  will  be  to  bring  about  a  common  temperature  of  ail 


HEAT.  191 

bodies.  When  this  common  temperature  has  been  attained,  no 
further  use  can  be  made  of  the  heat  in  those  bodies,  and  since, 
by  hypothesis,  all  other  available  energy  has  been  turned  into 
heat,  no  available  energy  will  be  left.  The  amount  of  energy 
in  the  universe  will  be  still  the  same,  but  none  of  it  can  be 
used.  The  process  by  which  this  final  condition  will  be  at- 
tained is  called  the  dissipation  of  energy.  . 

Boltzmann  has  explained  the  second  law  of  thermodynamics 
by  the  aid  of  the  kinetic  theory  of  heat.  On  the  assumption 
that  the  atoms  of  all  bodies  are  in  unordered  motion,  he  has 
shown  that  the  most  probable  condition  of  any  assemblage  of 
atoms  is  one  in  which  the  mean  kinetic  energy  of  each  atom  in 
the  same,  and  that,  when  this  condition  does  not  exist,  the 
probable  change  in  the  assemblage  is  toward  this  condition, 
and  not  away  from  it.  He  considers  that  the  second  law  of 
thermodynamics  is  the  experimental  equivalent  of  this  theo- 
retical conclusion,  and  expresses  the  general  tendency  of  bodies 
toward  this  most  probable  condition.  Boltzmann  suggests  a 
possible  escape  from  the  conclusion  that  the  final  condition  of 
the  universe  is  that  in  which  there  is  no  available  energy,  by 
calling  attention  to  the  fact  that,  although  the  general  ten- 
dency of  any  system  is  toward  its  most  probable  condition,  vet 
there  may  occur  in  the  system  such  a  combination  of  condi- 
tions that  for  a  while  the  tendency  will  be  toward  an  improb- 
able condition.  During  this  period  the  second  law  will  not 
hold  true,  and  at  the  end  of  it,  the  system  will  have  acquired 
a  store  of  available7  energy. 


192 


LIGHT. 

122.  General  Considerations  Respecting  Light. — The  sense 
of  sight  presents  to  us  a  large  number  of  phenomena  which 
we  associate  with  the  general  idea  of  light.     Our  object  is  to 
study  and  classify  these  phenomena,  and  so  far  as  possible  to 
explain  them.     We  shall  find  in  this  case,  for  the  first  time, 
that  a  sufficient  explanation  cannot  be  given  in  terms  of  the 
motion  of  matter,  and  that  it  will  be  necessary  to  assume  an 
additional  constituent  of  the  physical  universe  as  the  medium 
in  which  the  actions,  take  place  which  constitute  light.     We 
shall   find    further   that   this   medium   has   properties    unlike 
those  of  any  known  body  and  that  the  actions  in  it  do  not 
conform  to  the  laws  of  action  of  any  known  body.     Our  ex- 
planation therefore  consists  finally  in  the  reduction  of  all  the 
phenomena  of  light  to  actions  which  take  place  in  a  hypo- 
thetical- medium  according  to  certain  formal  laws. 

We  shall  find  it  expedient,  in  our  study  of  this  subject,  to 
examine  by  experiment  most  of  the  fundamental  and  im- 
portant facts  connected  with  it,  before  we  undertake  to  de- 
velop any  theory  of  light. 

123.  Origin  of  Light. — Opinions  were  divided,  in  antiquity, 
between   the   view   that   light    was   an   affection    of   external 
bodies,  passing  from  them  to  the  eye,  and  the  view  that  the 
eye  itself  was  the  source  of  light,  which  passed  out  from  it 
to  any  object  at  which  the  eye  was  looking.    The  latter  view 
was  unable  to  meet  the  criticism  of  Aristotle,  to  the  effect 
that,  if  it  were  true,  we  should  be  able  to  see  in  the  dark, 
except  by  artificial  additions  to  the  theory.     It  nevertheless 
held  a  prominent  place  in  thinking  for  many  years  after  that 
criticism  was  made  on  it.    Gradually,  however,  the  other  view 
displaced  it. 

The  eai'liest  observations  must  have  shown  that  light 
travels  in  straight  lines..  This  fact  is  so  well  known  that  we 
make  use  of  it  in  countless  ways,  without  considering  its  im- 
portance to  the  theory  of  light.  In  our  subsequent  discussions 


LIGHT.  193 

we  shall  use  the  word  ray  to  denote  any  straight  line  in  which 
light  travels. 

Another  observation,  which  was  made  so  early  that  no 
record  of  it  has  come  down  to  us,  is  that  of  the  reflection  of 
light.  When  light,  in  passing  from  an  object,  encounters  a 
smooth  surface,  it  is  more  or  less  completely  reflected  from 
that  surface.  If  we  select  a  point  on  the  reflecting  surface 
and  draw  from  it  a  perpendicular  to  that  surface,  and  if  we 
then  draw  from  the  source  of  light  the  ray  to  that  point,  this 
ray  will  make  with  the  normal  an  angle,  which  we  call  the 
angle  of  incidence.  The  light  from  the  source  is  reflected  at 
the  point  of  incidence  according  to  the  following  law:  The 
incident  and  reflected  rays  lie  in  a  plane  which  contains  the 
normal  to  the  reflecting  surface,  and  the  angles  of  incidence 
and  reflection  are  equal. 

A  third  observation,  which  for  the  present  we  shall  de- 
scribe only  in  general  terms,  is  that  of  the  refraction  of  light 
This  refraction  is  the  sudden  bending  or  change  of  direction 
which  a  ray  of  light  exhibits  as  it  passes  from  one  medium 
into  another.  It  is  refraction  which  makes  a  straight  stick 
look  bent  when  one  end  of  it  is  thrust  into  water. 

124.  Mirrors. — A  mirror  is,  in  general,  a  polished  surface 
used  for  the  reflection  of  light.  The  surface  usually  conforms 
very  nearly  to  some  geometrical  surface.  The  most  common 
mirrors  have  either  plane  or  spherical  surfaces. 

In  the  case  of  the  plane  mirror,  if  we  consider  a  point- 
source  of  light  set  up  in  front  of  it,  and  trace  the  course  of 
the  rays,  which  are  reflected  from  various  parts  of  it,  by 
means  of  the  general  law  of  reflection,  we  shall  find  that  they 
all  diverge  from  a  point  which  is  situated  behind  the  mirror 
on  the  perpendicular  drawn  from  the  source  to  the  mirror, 
and  as  far  behind  the  mirror  as  the  source  is  in  front  of  it. 
This  point  is  called  the  image  of  the  source. 

The  eye  sees  an  object  on  the  line  along  which  the  ray  of 
light  from  that  object  enters  the  eye.  If  we  consider  an  ex- 
tended object  placed  before  a  plane  mirror,  and  trace  the 
course  of  the  rays,  from  the  different  points  of  the  object, 


194  LIGHT. 

which  reach  the  eye  after  reflection  from  the  mirror,  we  shall 
find  that  they  proceed  from  images  of  those  points,  situated 
behind  the  mirror,  according  to  the  rule  just  given;  and  ac- 
cordingly, that  the  eye  perceives  in  the  mirror  a  collection  of 
images,  presenting  an  exact  counterpart  of  the  object,  as  it 
would  appear  if  it  were  transferred  to  the  place  of  its  image, 
with  the  exception  that  the  image  is  reversed,  the  right  hand 
of  the  object  appearing  to  be  the  left  hand  of  the  image,  and 
vice  versa. 

To  study  the  spherical  mirror,  we  suppose  a  small  portion 
of  a  spherical  shell  to  be  cut  off  by  a  plane  section.  The 
diameter  of  the  sphere  prolonged  through  the  middle  point  of 
the  section  we  call  the  axis.  If  reflection  occurs  at  the  inner 
surface  of  this  spherical  cap,  it  is  a  concave  mirror ;  if  at  the 
outer  surface,  a  convex  mirror.  The  point  where  the  axis  cuts 
the  mirror  is  called  the  vertex.  The  centre  of  the  sphere,  of 
which  the  mirror  forms  a  part,  is  the  centre  of  the  mirror, 
and  the  radius  of  that  sphere  is  the  radius  of  the  mirror. 

We  shall  consider  first  the  case  of  the  concave  mirror.  If 
we  suppose  the  source  of  light  to  be  a  point  on  the  axis  more 
distant  from  the  mirror  than  its  centre,  and  if  we  trace  the 
incident  and  reflected  rays  from  a  point  on  the  mirror,  which 
make  equal  angles  with  the  radius  drawn  to  that  point,  we 
find,  of  course,  that  the  reflected  rays  cut  the  axis  at  a  point 
lying  between  the  vertex  and  the  centre.  We  suppose  the 
mirror  to  be  small,  so  that  the  angle  between  the  incident 
ray  and  the  axis  is  always  small.  With  that  limitation  it 
may  be  proved  that  whatever  be  the  point  on  the  mirror  at 
which  the  incident  ray  meets  it,  the  reflected  ray  will  always 
pass  through  the  same  point  on  the  axis.  To  do  this  we  in- 
vestigate the  distance  from  the  vertex  of  th<*  point  on  the 
axis  at  which  the  reflected  ray  intersects  it.  Let  us  represent 
the  distance  of  the  source  from  the  centre  by  a,  the  distance  of 
the  point  of  intersection  from  the  centre  by  6.  Let  us  repre- 
sent further  the  radius  of  the  mirror  by  r,  and  the  distances 
from  a  point  on  the  mirror  to  the  source  and  the  point  of  inter- 
section respectively,  by  f  and  /'.  Because  the  angles  of  inci- 


LIGHT.  195 

dence  and  reflection  are  equal,  we  have  among  the  sides  of  the 
triangles  the  proportion  f:f'=a:b.  Now,  if  the  mirror  is 
small,  f  and  f  are  equal,  to  a  first  approximation,  to  the  dis- 
tances from  the  vertex  to  the  source  and  the  point  of  intersec- 
tion respectively.  We  have  therefore,  a=f — r,  and  b=r — f, 
and  using  these  values  in  the  proportion  already  given,  we 
obtain  fr-ff'=ff'—f'r.  Prom  this  we  obtain  (/  +  f')r  -  2ff', 
and  finally  \+j,=  \- 

This  equation  gives  the  distance  of  the  point  of  intersection 
from  the  vertex  in  terms  of  the  distance  of  the  source  from 
the  vertex  and  of  the  radius  of  the  mirror;  that  is,  in  terms  of 
quantities  which  are  independent  of  the  angle  between  the  inci- 
dent ray  and  the  axis.  The  same  point,  therefore,  will  be  a 
common  point  of  intersection  of  all  reflected  rays  coming  from 
the  given  source.  It  is  called  the  focus  of  the  source. 

It  is  plain  from  the  construction  that  the  source  and  focus 
may  be  interchanged.  The  two  points  thus  related  are  called 
conjugate  foci. 

If  the  source  is  at  an  infinite  distance  from  the  mirror,  the 
formula  shows  that  the  focus  will  be  midway  between  the 
centre  and  the  vertex.  The  point  thus  determined  is  called 
the  principal  focus  of  the  mirror.  As  the  source  moves  up 
toward  the  mirror  from  infinity,  the  focus  moves  toward  the 
centre.  When  the  source  reaches  the  centre,  it  coincides  with 
the  focus.  After  it  has  passed  the  centre,  the  focus  moves  out 
toward  infinity,  and  is  at  an  infinite  distance  when  the  source 
is  at  the  principal  focus.  During  all  these  changes  the  focus 
is  a  point  through  which  the  reflected  rays  actually  pass.  It 
is  therefore  called  a  real  focus. 

If  the  source  is  between  the  principal  focus  and  the  vertex, 
the  formula  shows  that  the  distance  of  the  focus  from  the 
vertex  is  negative.  This  indicates  that  the  focus  stands  be- 
hind the  mirror.  It  may  be  shown  by  construction  that,  in 
this  case,  the  reflected  rays  diverge  from  the  mirror.  They 
appear  to  come  from  a  point  behind  it.  This  point,  through 
which  the  rays  do  not  actually  pass,  is  called  a  virtual  focus. 


196  LIGHT. 

If  the  mirror  is  convex  instead  of  concave,  a  construction 
similar  to  the  one  already  made  will  show  that  the  reflected 
rays  always  diverge  from  the  mirror,  and  that  the  focus  of 
any  point  on  the  axis  is  a  virtual  focus.  A  demonstration 
similar  to  the  one  already  given  shows  that  a  formula  holds 
which  is  similar  to  the  one  already  obtained,  except  that  the 
terms  containing  lines  drawn  to  the  other  side  of  the  mirror 
have  the  minus  sign.  A  study  of  the  formula  shows  that  the 
principal  focus  lies  midway  between  the  centre  and  the  vertex, 
and  that  the  focus  of  any  real  point  on  the  axis  lies  between 
the  principal  focus  and  the  vertex. 

125.  Images. — If  a  small  extended  object  stands  transverse 
to  the  axis,  a  line  may  be  drawn  from  each  point  of  it  through 
the  centre  to  the  mirror,  which  will  be  an  axis  for  that  point, 
and  the  focus  of  the  point  will  lie  on  that  axis,  or  on  its  pro- 
longation through  the  mirror.     The  assemblage  of  foci  found 
in  this  way  for  the  different  points  of  the  object  constitutes 
what  is  called  an  image  of  the  object.     According  as  the  foci 
which  form  it  are  real  or  virtual,  the  image  is  called  a  real  or 
virtual  image.    Construction  shows,  in  the  case  of  the  concave 
mirror,  that,  so  long  as  the  object  lies  outside  the  principal 
focus,  the  image  is  real  and  inverted.     When  the  object  lies 
within  the  principal  focus,  the  image  is  virtual  and  erect.    In 
the  case  of  the  convex  mirror,  the  image  is  always  virtual  and 
erect. 

Real  images  can  always  be  studied  by  throwing  them  upon 
screens.  Virtual  images  can  only  be  examined  by  the  eye. 

126.  Mirrors  of  Large  Aperture. — The  solid  angle  subtended 
by  a  spherical  mirror  from  its  centre  is  called  its  aperture. 
The  mirrors  which  we  have  considered  hitherto,  and  to  which 
our  former  statements  apply,  were  of  very  small  aperture. 
When  the  aperture  becomes  considerable,  the  approximations 
upon   which  those   statements   depend   no   longer   hold   good. 
We  find  in  such  cases  that  no  definite  focus  exists.    The  rays 
reflected  from  the  different  points  of  the  mirror  cut  each  other 
in  such  a  way  as  to  form  a  specially  illuminated  surface.     In 
the    plane    diagram   this    becomes    a    special    line    called    the 
caustic. 


LIGHT.  197 

When  the  source  is  not  placed  on  the  axis  the  reflected 
light  passes  from  a  small  portion  of  the  mirror  in  such  a  way 
as  to  determine  a  line  in  space  through  which  the  rays  pass, 
and  which  is  in  consequence  specially  illuminated.  This  is 
called  the  first  focal  line.  It  is  at  right  angles  to  the  plane 
of  incidence.  After  passing  through  this  they  again  determine 
a  specially  illuminated  line,  which  is  in  the  plane  of  incidence. 
This  is  called  the  second  focal  line. 

127.  Refraction. — Attention  has  already  been  called  to  the 
fact  that  when  a  ray  of  light  passes  from  one  medium  into 
another,  it  is  refracted  or  bent  at  the  surface  which  separates 
the  media.  The  ray  in  each  medium  is  straight.  For  very 
many  years  the  attempts  which  were  made  to  discover  any  law 
governing  this  refraction  were  fruitless.  The  Greek  astronomer 
Ptolemy  was  the  first  to  investigate  this  question,  by  the  study 
of  the  refraction  between  air  and  water,  air  and  glass,  and 
water  and  glass.  He  could  find  no  general  law,  and  was  forced 
to  content  himself  with  an  empirical  table,  giving  the  angles 
of  refraction  corresponding  to  certain  angles  of  incidence. 
Even  Kepler,  who  investigated  the  same  question  by  the  help 
of  the  measurements  of  Ptolemy  and  others,  did  not  perceive 
the  true  law.  It  was  first  discovered  by  Snell  and  was  first 
published  by  Descartes.  It  may  be  stated  as  follows:  The 
incident  and  refracted  rays  lie  in  a  plane  which  contains  the 
normal  to  the  refracting  surface,  and  the  ratio  of  the  sine  of 
the  angle  of  incidence  to  the  sine  of  the  angle  of  refraction  is 
constant.  The  numerical  value  of  this  constant  depends  upon 
the  nature  of  the  media  which  lie  on  either  side  of  the  refract- 
ing surface.  When  the  media  are  specified,  both  as  to  their 
nature  and  as  to  the  order  in  which  they  are  considered,  the 
constant  is  called  the  index  of  refraction  from  the  first  to  the 
second.  Thus,  if  we  consider  the  incident  ray  in  water,  re- 
fracted at  a  surface  where  the  water  meets  a  block  of  glass, 
we  call  the  ratio  of  the  sines  of  the  angles  of  incidence  and 
refraction  in  this  case  the  index  of  refraction  from  water  to 
glass.  If  only  one  medium  is  specified,  it  is  assumed  that  it  is 
the  second  medium,  in  which  is  the  refracted  ray,  and  it  is 


198  LIGHT. 

assumed  that  the  first  medium  is  either  air,  or  better,  vacuum. 
The  constant  in  this  case  is  called  the  index  of  refraction  of 
the  second  medium.  For  example,  when  light  passes  from  air 
into  water,  the  ratio  of  the  sines  is  1.333.  This  number  is 
called  the  index  of  refraction  of  water. 

Experiment  shows  that,  for  all  substances  with  which  we 
are  at  present  concerned,  the  index  of  refraction  is  greater 
than  unity,  so  that  the  angle  of  refraction  is  less  than  the 
angle  of  incidence.  If  the  index  of  refraction  between  two 
media  is  greater  than  unity,  the  second  medium,  in  which  is 
the  refracted  ray,  is  said  to  be  optically  denser  than  the  other. 
The  term  is  a  convenient  one,  if  we  are  careful  not  to  take  it 
to  mean  the  actual  density  of  the  medium. 

If  the  refracted  ray  is  turned  back  on  itself  by  reflection 
in  a  plane  mirror,  or  if  a  source  of  light  is  placed  in  the  line 
of  the  refracted  ray,  the  ray  which  proceeds  through  the 
second  medium  will  be  refracted  at  the  original  point  of  inci- 
dence, and  the  refracted  ray  thus  obtained  will  coincide  with 
the  original  incident  ray.  The  index  of  refraction  between 
two  media  when  they  are  taken  in  one  order,  is  therefore  the 
reciprocal  of  the  index  of  refraction  when  they  are  taken  in 
the  reverse  order. 

Let  us  consider  for  a  moment  the  simple  case  of  refraction 
between  air  and  water,  which  is  a  typical  one.  When  the  inci- 
dence is  perpendicular,  the  direction  of  the  refracted  ray  is 
the  same  as  that  of  the  incident  ray.  As  the  angle  of  inci- 
dence, in  the  air,  increases,  the  angle  of  refraction,  in  the 
water,  increases  also,  though  not  so  rapidly.  The  ratio  of  the 
sines  of  the  two  angles  is  always  the  same  number.  When  the 
angle  of  incidence  becomes  a  right  angle,  which  is  as  large  as 
it  can  be,  the  angle  of  refraction  is  less  than  a  right  angle, 
and  the  sine  'of  the  angle  of  refraction  equals  the  reciprocal 
of  the  index  of  refraction.  Consider  next  the  refraction  of  a 
ray  which  is  incident  in  water  on  the  surface  separating  the 
water  from  the  air.  When  the  incident  ray  is  perpendicular 
to  that  surface,  the  direction  of  the  refracted  ray  is  un- 
changed. As  the  angle  of  incidence,  in  the  water,  increases, 


the  angle  of  refraction,  in  the  ah-,  also  increases,  but  more 
rapidly.  The  ratio  of  the  sines  of  the  two  angles  is  the  re- 
ciprocal of  the  index  of  refraction  of  water.  When  the  angle 
of  incidence  is  so  great  that  its  sine  is  equal  to  the  reciprocal 
of  the  index  of  refraction,  the  sine  of  the  angle  of  refraction 
is  equal  to  unity,  and  the  angle  of  refraction  is  a  right  angle. 
The  refracted  ray,  in  this  case,  just  emerges  from  the  water 
into  the  air.  If  the  angle  of  incidence  is  made  still  greater,  it 
follows  from  the  formula  that  the  sine  of  the  angle  of  re- 
fraction is  greater  than  unity.  This  impossible  result  indi- 
cates a  failure  of  the  law  of  refraction.  \n  fact,  after  the 
angle  of  incidence  has  passed  the  limiting  value,  for  which  its 
sine  is  equal  to  the  reciprocal  of  the  index  of  refraction,  light 
no  longer  emerges  into  the  air.  This  limiting  value  of  the 
angle  of  incidence  is  called  the  critical  angle.  Light  incident  at 
an  angle  which  is  greater  than  the  critical  angle  is  totally 
reflected  within  the  water. 

128.  Fermat's  Law  of  Least  Time. — A  general  law  was 
announced  by  Fermat,  which  governs  both  reflection  and  re- 
fraction. It  may  be  stated  by  saying,  that  the  time  taken  by 
light  to  pass  from  one  point  to  another  by  way  of  a  reflecting 
or  refracting  surface  is  a  minimum.  To  illustrate  this  for  the 
case  of  reflection,  let  us  consider  two  points  on  the  same  side 
of  a  reflecting  surface,  from  one  of  which  the  incident  ray 
starts.  The  other  point  is  the  one  through  which  the  reflected 
ray  passes.  If  we  select  any  point  on  the  reflecting  surface, 
and  draw  lines  from  it  to  the  two  points  already  chosen,  it 
may  be  shown  that  the  sum  of  the  lengths  of  these  lines  is  a 
minimum  when  the  point  on  the  surface  is  so  placed  that  the 
lines  drawn  to  it  from  the  other  points  lie  in  a  plane  contain- 
ing the  normal  to  the  surface  and  make  equal  angles  with  that 
normal.  The  lines  whose  lengths  fulfill  the  condition  that 
their  sum  is  a  minimum  thus  conform  to  the  law  of  reflection, 
and  represent  the  incident  and  reflected  rays  which  pass 
through  the  two  points.  If  light  travels  in  a  particular 
medium  with  a  definite  velocity,  the  time  required  for  it  to 
pass  from  the  one  point  to  the  other  by  the  lines  which  fulfill 
the  minimum  conditions  is  the  least  possible. 


'200  LIGHT. 

In  the  case  of  refraction  Fermat  assumed  that  the  rate  at 
which  light  travels  is  different  in  different  media,  and  that 
the  ratio  of  the  velocities  in  any  two  media  is  equal  to  the 
index  of  refraction  between  those  media.  If  we  select  two 
points,  one  to  serve  as  the  source  from  which  the  incident 
ray  comes,  the  other  to  serve  as  the  point  through  which  the 
refracted  ray  passes,  it  may  be  shown  that  this  law  of  the 
velocity  leads  to  the  conclusion  that  the  path  along  which 
light  will  pass  from  the  one  point  to  the  other  in  the  least 
time  is  that  which  conforms  to  the  law  of  refraction. 

This  general  principle  is  known  as  the  principle  of  least 
time.  As  announced  by  Fermat  it  was  simply  an  hypothesis, 
for  which  no  experimental  proof  or  even  theoretical  argument 
could  be  given.  It  was  subsequently  shown  to  be  a  conse- 
quence of  the  wave  theory  of  light,  and  may  now  be  used  with 
confidence  in  the  solution  of  problems. 

129.  Prisms. — A  block  of  any  transparent  substance,  en- 
closed between  two  planes  which  meet  at  an  edge,  is  called  a 
prism.  The  angle  between  the  planes  is  called  the  refracting 
angle  of  the  prism.  We  shall  consider  only  the  simple  case  in 
which  the  substance  of  the  prism  is  glass,  and  the  surrounding 
medium  air.  If  a  ray  of  light  is  incident  obliquely  upon  one 
face  of  this  prism,  it  is  refracted  in  the  glass  toward  the  nor- 
mal to  the  first  surface,  and  travels  on  in  the  glass  until  it 
meets  the  second  surface.  There  it  is  again  refracted,  this  time 
away  from  the  normal  to  the  second  surface.  The  result  of 
these  two  refractions,  at  least  for  many  cases  of  incidence,  is 
that  the  two  refractions  combine  to  make  the  emerging  refract- 
ed ray  deviate  from  the  original  direction  of  the  incident  ray, 
toward  the  base  of  the  prism.  Of  course,  if  the  original  inci- 
dence is  such  that  the  refracted  ray  in  the  glass  meets  the 
second  surface  so  that  its  angle  of  incidence  there  is  greater 
than  the  critical  angle,  the  ray  will  not  emerge  from  the  prism. 
Analysis  shows  that  the  deviation  of  the  emergent  ray  from 
the  direction  of  the  incident  ray  is  least  when  the  refracted 
ray  in  the  glass  is  at  right  angles  to  the  line  which  bisects 
the  refracting  angle  of  the  prism.  When  the  prism  is  so  placed 


LIOHT.  201 

that  this  condition  obtains,  it  is  said  to  be  in  the  position  of 
minimum  deviation.  As  a  prism  is  usually  a  block  whose 
cross-section  is  a  triangle,  with  its  base  perpendicular  to  the 
line  bisecting  the  refracting  angle,  the  condition  of  minimum 
deviation  is  often  described  by  saying  that  in  it  the  refracted 
ray  in  the  glass  is  parallel  with  the  base  of  the  prism. 

130.  Refraction  at  a  Single  Spherical  Surface. — If  light  is 
incident  from  a  point  in  air  upon  a  small  portion  of  a  spherical 
surface  bounding  another  medium,  so  placed  that  one  of  the 
rays  from  the  source  meets  that  surface  perpendicularly,   it 
may  be  shown  that  all  the  refracted  rays  will  pass  through  a 
point  on  that  perpendicular  line  or  on  it  produced  into  the 
second  medium.    This  point  is  a  focus.    If  the  focus  is  in  the 
second  medium,  the  refracted  rays  actually  pass   through  it, 
and  it  is  a  real  focus.     If  it  is  in  the  air,  the  refracted  rays  do 
not  pass  through  it,  but  appear  to  diverge  from  it,  and  it  is  a 
virtual  focus.     The  position  of  the  focus  depends  not  only  on 
the  position  of  the  source,  and  on  the  radius  of  the  spherical 
surface,  but  also  on  the  index  of  refraction  of  the  medium. 

As  a  typical  case  we  consider  the  spherical  surface  concave 
toward  the  source.  In  this  case  we  may  show  by  construction 
that  the  focus  lies  in  the  air,  or  is  a  virtual  focus.  The  form- 
ula connecting  the  distance  f  of  the  focus  from  the  surface 
with  the  distance  f  of  the  source  from  the  surface,  with  the 

radius   r,   and    with   the   index   of  refraction  n,  is  ~ —  =nl-  • 

In  this  case  all  the  lines  are  measured  from  the  surface  toward 
the  source  of  light. 

If  the  surface  is  convex  toward  the  source,  a  construction 
shows  that  the  focus  may  sometimes  he  real.  The  formula 
for  this  case  is  obtained  from  the  one  just  given  by  changing 
the  signs  of  those  lines  in  it  which  are  drawn  from  the  surface 
away  from  the  source  of  light. 

131.  Lenses. — A  transparent  body  which  is  bounded  by  two 
spherical  surfaces  is  called  a  lens.    A  line  drawn  through  the 
lens,  perpendicular  to  both  its  bounding  surfaces,  may  be  called 
its  axis.    If  the  source  of  light  is  placed  on  the  axis,  refraction 
will  occur  at  the  first  surface  and  the  refracted  rays  will  pro- 


202  LIGHT 

ceed.  as  if  they  were  coming  from  or  going  toward  the  focus 
formed  by  this  surface.  When  they  meet  the  second  surface, 
they  will  again  be  refracted,  and  their  new  directions  will  de- 
termine a  second  focus.  This  focus  is  the  focus  of  the  source 
formed  by  the  lens.  Its  position  may  be  found,  as  our  descrip- 
tion lias  indicated,  by  using  twice  the  formula  for  the  focus 
due  to  a  single  spherical  surface. 

There  are  two  general  types  of  lenses,  called  respectively 
convex  and  concave  lenses.  In  convex  lenses  the  surfaces 
which  bound  them  are  so  shaped  that  the  lens  is  thickest  in 
the  middle;  in  concave  lenses,  the  lens  is  thinnest  in  the 
middle.  It  may  be  shown  by  construction  and  proved  by 
analysis,  that  the  effect  of  the  convex  lens  is  to  cause  the  rays 
which  fall  upon  it  to  converge,  or  at  least  to  diverge  less 
widely  than  before  they  met  the  lens.  On  the  other  hand,  the 
effect  of  a  concave  lens  is  to  diverge  the  rays  more  widely. 
These  classes  of  lenses  are,  therefore,  called,  respectively,  con- 
verging and  diverging  lenses. 

The  typical  lens  is  the  convex  meniscus.  The  radii  of  the 
two  surfaces  which  bound  it  are  both  drawn  toward  the  source 
of  light,  and  the  radius  of  the  second  surface  is  less  than  that 
of  the  first.  If  we  suppose  the  lens  to  be  so  thin  that  we  can 
•neglect  its  thickness  in  comparison  with  the  other  distances 
involved,  we  may  obtain  a  simple  formula  for  the  distance  f 
of  the  focus  from  the  lens  in  terms  of  the  distance  f  of  the 
source  from  the  lens,  of  the  radii  r  and  s  of  the  first  and  sec- 
ond surfaces  respectively,  and  of  the  index  of  refraction  n  of 

the  substance  of  the  lens.  This  formula  is  1J—1e=^(n—l)(1  —~\. 

\r       s  I 

In  this  formula  all  the  lines  are  measured  from  the  lens 
toward  the  source.  To  adapt  it  to  any  other  form  of  lens  we 
change  the  signs  of  those  lines  which  are  drawn  from  the  lens 
away  from  the  source  of  light.  The  sign  of  f  is  left  unchanged, 
this  quantity  being  the  unknown  term  in  the  formula.  If  for 
given  conditions  its  sign  is  positive,  the  focus  is  on  the  same 
side  of  the  lens  as  the  source  of  light,  and  is  therefore  a  vir- 
tual focus.  If  it  is  negative,  the  focus  lies  on  the  other  side 
of  the  lens  and  is  a  real  focus. 


I.KIIIT.  203 

If  the  source  of  light  is  at  an  infinite  distance,  the  focus 
obtained  is  called  the  principal  focus,  and  its  distance  from 
the  lens,  the  focal  length  of  the  lens.  It  is  plain  that  the 
source  and  focus  may  be  interchanged.  The  points  thus  related 
are  therefore  called  conjugate  foci. 

We  may  illustrate  the  use  of  the  formula  by  applying  it  to 
the  double  convex  lens,  the  ordinary  burning  glass.  In  this 
lens  we  must  set  the  radius  of  the  first  surface  negative,  since 
it  is  drawn  away  from  the  source  of  light.  When  the  source 
is  sit  infinity  the  focal  length  of  the  lens  is  the  reciprocal  of 

—  (n  —  1)  I  -  +  -  I   and  is  negative,  so  that  light  falling  from 

an  infinite  distance  on  such  a  lens  converges  to  a  real  focus. 
As  the  source  moves  up  toward  the  lens,  the  focus  moves  away 
from  it,  until,  when  the  source  is  at  a  distance  from  the  lens 
equal  to  its  focal  length,  the  focus  is  at  an  infinite  distance 
from  it.  If  the  source  moves  still  nearer  to  the  lens,  the  sign 
of  f  becomes  positive,  showing  that  the  focus  is  on  the  same 
side  of  the  lens  as  the  source,  or  is  now  a  virtual  focus. 

132.  Images. — If  a  small  extended  object  stands  transverse 
to  the  axis  of  the  lens,  the  light  coming  from  any  point  of  it 
will  come  to  a  focus  on  the  line  drawn  from  it  through  the 
centre  of  the  lens.     The  assemblage  of  foci  found  in  this  way 
for  the  different  points  of  the  object  is  its  image.     By  con- 
struction we  may  show  that  real  images  are  inverted,  while 
virtual  images  are  erect.    The  ratio  between  the  height  of  the 
object  and  the  height  of  the  image  is  equal  to  the  ratio  be- 
tween their  respective  distances  from  the  lens. 

133.  The  Camera.     The  Eye. — If  a  double  convex  lens  is 
placed   in   an   aperture   made   in   the   wall    of   a   chamber   or 
camera,  images  of  external  objects  will  be  brought  to  a  focus, 
with   more  or   less   distinctness,    on   the   opposite   wall.     By 
using  a  movable  screen,  the  image  of  any  particular  object  can 
be  brought  to  exact  focus  on  it.    Such  a  screen  is  used  in  the 
ordinary    photographic    camera.      If    the    screen    cannot    be 
moved,  as  in  some  forms  of  camera,  the  lens  is  so  selected  as 
to  give  sharp  images  of  objects  at  a  certain  distance.     The 


204  LIGHT. 

images  of  objects  at  a  less  distance  are  not  sharp,  and  clear 
pictures  of  them  cannot  be  obtained.  The  images  of  more 
distant  objects  are  also  not  sharp,  but  the  indistinctness  in 
them  is  much  less  marked.  It  is  plain  that,  if  we  had  at  our 
disposal  a  number  of  lenses  of  different  curvatures,  or  if  we 
were  able  to  modify  the  curvature  of  our  lens,  we  might 
obtain  a  distinct  image  of  any  object,  even  on  a  fixed  screen. 
.The  eye  is  essentially  a  camera  of  this  sort.  Light  which 
enters  through  the  pupil  falls  upon  the  crystalline  lens,  and 
is  brought  to  a  focus  upon  the  retina.  The  optic  nerves  are 
distributed  over  the  retina,  and  are  affected  by  the  image 
which  falls  upon  them.  By  a  special  muscle  which  controls 
the  crystalline  lens,  its  shape  is  modified  in  order  to  produce 
a  sharp  image,  on  the  retina,  of  the  particular  object  which  is 
under  observation. 

The  angle  subtended  at  the  eye  by  the  rays  coming  from 
the  extreme  points  of  an  object  is  called  the  visual  angle  sub- 
tended by  that  object.  It  is  evident  that  the  perception  of 
small  details  depends  upon  the  visual  angles  subtended  by 
them,  and  hence  that  they  will  be  better  perceived  the  nearer 
they  are  to  the  eye.  We  cannot,  however,  push  the  examin- 
ation of  details  to  an  extreme  by  bringing  the  object  very 
near  the  eye;  for,  as  the  object  is  brought  nearer,  its  image 
tends  to  recede  behind  the  retina,  and  it  can  only  be  kept  on 
the  retina  by  increasing  the  sphericity  of  the  lens.  This  can- 
not be  done  beyond  certain  limits,  and  after  the  object  has 
approached  the  eye  within  a  certain  distance,  it  can  no  longer 
be  seen  distinctly.  Even  when  it  can  be  seen  distinctly,  the 
effort  involved  soon  tires  the  eye.  The  normal  nearest  dis- 
tance at  which  objects  can  be  seen  distinctly  without  per- 
ceptible effort  is  called  the  distance  of  distinct  vision. 

The  nearsighted  eye  is  one  in  which  the  crystalline  lens 
cannot  be  flattened  sufficiently.  Because  of  its  too  great 
sphericity,  it  throws  the  images  of  distant  objects  in  front  of 
the  retina.  This  defect  is  corrected  by  placing  before  the  eye 
a  concave  lens,  which  slightly  increases  the  divergence  of  the 
light  coming  from  distant  objects,  and  so  throws  back  their 


LIGHT.  205 

images  on  the  retina.  The  far-sighted  eye  is  one  in  which  the 
crystalline  lens  cannot  be  made  sufficiently  curved.  Because 
of  its  flatness,  it  throws  the  images  of  near  objects  behind  the 
retina.  This  defect  is  corrected  by  placing  before  the  eye  a 
convex  lens,  which  lessen*  the  divergence  of  the  liijht  coming  from 
near  objects,  so  that  distinct  images  of  them  are  cast  on  the 
retina. 

134.  Optical  Instruments. — By  viewing  an  object  through 
a  lens  or  a  combination  of  lenses,  its  visual  angle  may  be  in- 
creased, while  at  the  same  time  the  light  coming  from  it  can 
be  properly  brought  to  a  focus  on  the  retina  of  the  eye.  Such 
combinations  are  called  optical  instruments.  In  general  they 
may  be  said  to  be  used  for  magnifying  objects.  The  ratio  of 
the  visual  angle  of  the  object  when  seen  through  the  instru- 
ment to  its  visual  angle  when  seen  with  the  unaided  eye 
may  be  taken  as  a  measure  of  the  magnifying  power  of  the 
instrument. 

The  convex  lens  is  used  as  a  magnifying  glass.  If  an  object 
which  we  wish  to  examine  is  placed  in  front  of  the  lens,  at  a 
distance  from  it  somewhat  less  than  its  focal  length,  the  rays 
from  the  object  will  be  rendered  only  slightly  divergent  by  a 
passage  through  the  lens,  so  that  an  eye  which  receives  them 
after  they  have  passed  the  lens  can  bring  them  to  a  focus  on 
the  retina.  These  rays  appear  to  the  eye  to  come  from  the 
virtual  image  of  the  object.  By  properly  adjusting  the  dis- 
tance between  the  lens  and  the  object,  this  virtual  image  may 
be  brought  to  the  position  of  distinct  vision.  When  examined 
by  the  eye  it  appears  erect  and  larger  than  the  object.  The 
amount  of  magnification  depends  on  the  shape  of  the  lens.  It 
increases  as  the  focal  length  of  the  lens  decreases. 

The  telescope,  in  its  first  form,  was  constructed  by  a  Dutch 
optician,  Jansen,  who  was  led  to  it  by  an  accidental  observa- 
tion made  by  his  children.  In  the  same  form  it  was  invented 
by  Galileo,  and  is  generally  known  by  his  name.  The  opera 
glass  is  an  example  of  this  form  of  telescope.  It  consists 
essentially  of  a  convex  lens,  or  object  glass,  and  a  concave 
lens,  or  eye  piece,  carried  in  a  tube  whose  length 'may  be  ad- 


20(5  LIGHT. 

justed.  The  distance  between  the  lenses  is  less  than  the  focal 
length  of  the  object  glass.  When  the  rays  from  a  point  of  a 
distant  object  pass  through  the  object  glass,  they  are  made 
to  converge.  Their  convergence  is  so  rapid  that  the  unaided 
eye  cannot  bring  them  to  a  focus  on  the  retina.  But,  by 
passage  through  the  concave  lens,  they  are  made  to  diverge 
slightly,  so  that  the  eye  looking  toward  that  lens  is  able  to 
bring  them  to  a  focus  on  the  retina.  The  light  therefore 
enters  the  eye  as  if  it  came  from  a  point  situated  at  the  apex 
of  the  cone  of  rays  which  enters  the  pupil  from  the  real  point 
of  the  object,  after  passing  through  the  lenses.  The  eye 
therefor*  sees  an  image  of  the  object.  By  a  suitable  adjust- 
ment of  the  distance  between  the  lenses,  the  image,  which  i? 
virtual  and  erect,  may  be  set  at  any  apparent  distance  desired. 
The  construction  will  show  that  it  subtends  a  larger  visual 
angle  than  the  object  itself. 

The  astronomical  telescope,  in  its  simplest  form,  contains  a 
convex  lens,  as  object  glass,  and  a  convex  lens,  as  eye  piece. 
The  rays  from  a  distant  object  form  a  real  inverted  image  of 
it  by  passage  through  the  object  glass.  After  passing  through 
the  points  of  tht  image,  they  proceed  as  diverging  rays.  If 
they  are  then  intercepted  by  the  second  convex  lens  or  eye 
piece,  they  will  behave  exactly  as  the  rays  do  which  come 
from  the  points  of  an  object  that  is  examined  by  the  magnify- 
ing glass.  On  looking  toward  the  eye  piece,  the  eye  will  see  a 
virtual  image  of  the  real  image  formed  by  the  object  glass. 
The  visual  angle  subtended  by  this  image  is  greater  than  that 
subtended  by  the  object.  Since  the  real  image  examined  by 
the  eye  piece  is  inverted  as  respects  the  object,  the  virtual 
image  seen  is  also  inverted.  This  circumstance  is  of  no  conse- 
quence in  astronomical  observation.  In  instruments  of  this 
sort,  designed  for  terrestrial  observation,  a  pair  of  lenses  is  inter- 
posed between  the  other  two,  by  which  the  real  image  is 
inverted  so  as  to  be  erect.  The  virtual  image  seen  is  then 
also  erect. 

The  microscope  is  also  a  combination  of  two  convex  lenses. 
The  object  examined  by  it  is  placed  just  outside  the  principal 


LIGHT.  207 

focus  of  the  object  glass,  and  the  enlarged  inverted  image  of 
it,  formed  by  the  object  glass,  is  examined  by  the  convex  lens 
forming  the  eye  piece.  The  magnified  image  in  this  case  is 
inverted. 

135.  Lenses  of  Large  Aperture. — The  formulae  which  have 
been  given  and  the  statements  which  have  been  made,  with 
respect  to  lenses,  hold  true  when  only  a  small  portion  of  the 
lens  around  its  axis  is  used,  and  then  as  a  first  approximation. 
When   lenses   of   large   aperture   are   used,   the   light   from   a 
source   on  the  axis  does  not  come  to  an  exact   focus.     The 
intersection    of    the    various    rays    determines    an    especially 
illuminated  surface,  called  the  caustic  surface. 

Furthermore,  in  the  case  of  such  lenses,  the  dimensions  of 
the  image  are  not  in  the  same  proportion  to  each  other  as 
those  of  the  object.  These  defects  of  the  image  are  said  to 
be  due  to  the  spherical  aberration  of  the  lens.  By  combining 
two  lenses,  so  shaped  that  the  spherical  aberration  of  one  is 
in  the  opposite  sense  to  that  of  the  other,  a  single  object  glass 
or  eye  piece  can  be  constructed  which  is  free  from  spherical 
aberration. 

136.  Intensity  of  Light. — By  the  intensity  of  light  is  meant 
its  illuminating  power,  as  judged  by  the  eye  on  observation 
of  the  illumination  of  some  standard  surface.    It  is  measured, 
or   rather   two   intensities   are   compared,   by   an   instrument 
called  the  photometer.     One  of  the  earliest  forms  of  photo- 
meter, invented  by  Count  Rumford,  is  made  by  setting  up  a 
vertical   rod  at  a  little  distance   from  a  white   screen.     The 
two  sources  of  light  to  be  compared  are  set  so  as  to  cast 
shadows  of  the  rod  near  each  other  on  the  screen.    The  space 
covered  by  the  shadow  from  one  source  is  illuminated  by  the 
light  from  the  other  source.     The  sources  are  then  moved 
about  until  the  two  shadows  appear  equally  illuminated,  and 
it  is  then  said  that  the  intensity  of  the  light  from  the  two 
sources  is  the  same.    By  comparing  the  effects  of  the  sources 
when  they  are  set  at  different  distances  from  the  screen,  it  is 
found  that  the  intensity  of  the  light  from  a   source  varies 
inversely  with  the  square  of  the  distance  from  the  source. 


208  LIGHT. 

137.  The  Velocity  of  Light. — In  our  work,  up  to  this  time, 
we  have  treated  the  rays  of  light  as  if  they  marked  paths 
along  which  light  travels  from  its  source.  No  proof  has  been 
given  to  justify  this,  nor  is  any  needed,  so  long  as  we  adhere 
to  our  original  conception  of  light,  as  being  something  which 
originates  at  external  bodies  and  reaches  the  eye  from  them. 
Very  different  ideas  prevailed  among  the  older  students  of 
optics  about  the  velocity  with  which  light  travels.  Descartes 
thought  that  its  velocity  was  infinitely  great.  Galileo,  on 
the  other  hand,  conceived  of  it  as  being  possibly  so  small  that 
it  could  be  detected  by  a  simple  experiment.  He  proposed  that 
two  observers,  furnished  with  lanterns,  should  occupy  two 
stations  at  a  considerable  distance  from  each  other.  The  first 
observer  was  to  expose  his  lantern.  The  second  observer,  see- 
ing the  light  from  it,  was  to  expose  his  lantern  in  turn,  and 
the  light  from  it  was  to  be  observed  by  the  observer  at  the 
first  station.  From  the  time  which  elapsed  between  the  ex- 
posure of  the  first  lantern  and  the  reception  of  light  from  the 
second,  Galileo  hoped  to  determine  the  velocity  of  light. 
When  the  experiment  was  tried,  it  was  found  that  no  per- 
ceptible time  elapsed  between  these  two  events,  except  that 
which  was  unavoidably  taken  by  the  second  observer  in 
receiving  the  sensation  of  light  from  the  first  lantern  and  in 
exposing  his  own.  So  far  as  this  experiment  went,  the  velocity 
of  light  was  infinite. 

It  was  proved  not  to  be  infinite,  and  its  value  was  deter- 
mined, by  the  Danish  astronomer  Roemer,  in  1676.  Roemer 
had  been  engaged  in  studying  the  revolutions  of  Jupiter's 
satellites.  In  the  course  of  their  revolutions,  these  satellites 
are  often  eclipsed  by  passing  into  Jupiter's  shadow,  and  the 
times  of  these  eclipses  can  be  very  exactly  noted.  From  a 
succession  of  such  observations,  Roemer  found  the  period  of 
revolution  of  one  of  the  satellites,  and  predicted  the  moments 
of  its  eclipses.  On  continuing  his  observations,  he  found  that 
the  actual  times  of  the  eclipses  gradually  departed  from  the 
predicted  times.  When  the  observations  were  analyzed,  it  was 
found  that,  when  the  earth  was  moving  away  from  Jupiter, 


LIGHT.  209 

the  period  of  the  satellite  appeared  longer,  and  when  the  earth 
was  moving  toward  Jupiter,  shorter,  than  the  mean  or  average 
value  of  all  the  observed  periods.  Thus,  if  the  times  of  eclipse 
were  predicted  from  observations  made  when  the  earth  was 
nearest  Jupiter,  the  observed  times  when  the  earth  was  far- 
thest from  Jupiter  would  lag  behind  the  predicted  ones.  Roe- 
iner  concluded  that  these  observations  could  be  best  explained 
by  supposing  that  light  travels  with  a  finite,  though  very  great, 
velocity.  On  the  basis  of  his  observations,  he  stated  that  light 
took  22  minutes  to  travel  over  the  diameter  of  the  earth's 
orbit. 

This  conclusion  of  Roemer's  was  for  many  years  uncon- 
firmed. At  length,  in  1729,  the  English  astronomer  Bradley 
made  a  discovery  which  seemed  to  confirm  it.  From  long  con- 
tinued observations  of  the  positions  of  the  fixed  stars,  he  found 
that  the  stars  apparently  describe  small  paths  or  orbits,  which 
are  completed  in  one  year.  The  paths  which  those  stars  de- 
scribe which  lie  near  the  plane  of  the  earth's  orbit  are  short 
straight  lines.  Those  of  the  stars  which  lie  farther  away  from 
this  plane  are  ellipses,  which  approach  circles  more  nearly  as 
the  star  lies  farther  from  that  plane.  In  Bradley's  time  it 
was  commonly  thought  that  light  consisted  of  streams  of 
minute  particles,  .shot  out  in  straight  lines  from  the  luminous 
body.  On  this  theory  of  light,  it  was  easy  to  explain  Bradley's 
observation.  For,  the  velocity  of  the  light  particles  entering 
the  eye  would  be  the  resultant  of  the  velocity  of  light  in  space, 
and  of  a  velocity  equal  to  that  of  the  earth  in  its  orbit  and  in 
the  opposite  direction:  and  the  direction  in  which  the  light 
particle  would  reach  the  eye  would  be  the  direction  of  this 
resultant.  When  the  earth  was  moving  across  the  path  of  the 
light,  the  direction  of  this  resultant  would  be  such  that  the 
star  would  appear  displaced  from  its  true  position  in  the  direc- 
tion of  the  earth's  motion.  From  the  amount  of  the  deviation 
and  the  known  velocity  of  the  earth  in  its  orbit,  the  velocity  of 
light  could  be  calculated.  The  result  obtained  was  in  good 
agreement  with  that  previously  obtained  by  Roemer's  method 
as  improved  by  later  observers.  The  phenomenon  discovered  by 
Bradley  is  called  the  aberration  of  light. 


210  LIGHT. 

In  subsequent  years  the  velocity  of  light  has  frequently 
been  measured  by  methods  which  do  not  involve  astronomical 
observations,  and  the  values  obtained  by  Roemer's  and  Brad- 
ley's  methods  are  confirmed.  The  methods  employed  for  this 
purpose  will  subsequently  be  discussed.  While  Bradley's  con- 
clusion was  thus  confirmed,  the  particular  argument  by  which 
he  reached  it  has  long  ago  been  discarded,  because  of  the  aban- 
donment of  the  theory  upon  which  that  argument  was  based. 
The  explanation  of  aberration  by  means  of  the  theory  of  light 
now  universally  accepted  is  still  under  discussion.  There  can 
be  no  doubt  that  the  aberration  depends  on  the  velocity  of 
light  in  the  way  Bradley  assumed  it  did,  but  it  is  extremely 
difficult  to  reconcile  aberration  with  other  results  of  experi- 
ment. 

138.  Composition  of  White  Light. — The  rainbow  naturally 
attracted  the  attention  of  the  early  students  of  optics.  It  was 
soon  perceived  that  the  light  which  reaches  the  eye  from  the 
bow  is  light  from  the  sun,  which  has  been  reflected  by  the  drops 
of  rain.  Descartes  was  able  to  explain  the  size  of  the  bow  by 
showing  that,  though  light  will  reach  the  eye  from  drops  in 
any  position  after  one  internal  reflection,  yet  that  the  light 
that  comes  from  those  drops  situated  in  the  direction  from 
the  eye  in  which  the  bow  appears  to  be,  will  be  much  less 
widely  divergent,  or,  as  he  expressed  it,  will  be  denser,  than 
the  light  from  drops  in  other  directions.  The  colors  of  the 
bow  were  thought  t<>  h<>  sw«ndnry  ph^noim-mi.  ISiniiliir  colors 
were  observed  in  light  reflected  from  drops  of  water,  and  in 
the  light  which  had  passed  through  a  prism  or  a  lens.  White 
light  was  supposed  to  be  the  simplest  sort  of  light,  and  the 
colors  produced  by  passage  through  bodies  were  supposed  to  be 
caused  by  modifications  impressed  by  those  bodies  upon  white 
light. 

It  was  shown  by  Newton,  in  1672,  that  this  conception  of 
white  light  was  entirely  erroneous.  Newton  received  a  beam 
of  sunlight,  which  had  passed  through  a  small  hole  in  a  win- 
dow shutter,  upon  a  prism,  and  cast  the  emergent  light  on  a 
screen.  He  found  that  the  light  on  the  screen,  instead  of  being 


LIGHT.  211 

a  circle,  was  a  long  colored  strip.  Some  of  the  colors  were 
more  deviated  than  others.  The  colored  strip  he  called  the 
spectrum.  The  colors  noted  by  him,  beginning  with  the  one 
which  is  least  deviated,  were  red,  orange,  yellow,  green,  blue, 
indigo,  and  violet.  We  may  interpret  this  experiment  by  say- 
ing that  the  index  of  refraction  of  glass  for  these  different 
colors  is  different,  and  that  white  light,  instead  of  being  simple, 
contains  all  these  colors,  and  is  analyzed  into  them  by  the 
prism  because  of  their  different  refrangibilities.  Newton 
proved  that  the  refractive  index  was  different  for  the  different 
colors,  by  placing  another  prism  behind  a  small  hole  cut  in  the 
screen  upon  which  the  spectrum  was  cast,  and  by  turning  the 
first  prism  so  as  to  throw  the  different  colors  successively 
through  the  hole.  When  this  was  done,  each  color  in  turn  was 
refracted,  without  any  modification  of  color,  and  the  deviation 
was  different  for  each.  To  confirm  this  conclusion  Newton 
superposed  the  colors  of  the  spectrum  by  the  aid  of  other 
prisms  or  of  mirrors,  and  in  other  ways,  and  found  that  in 
each  case  white  light  was  again  obtained. 

As  we  now  use  the  prism,  white  light  is  passed  through  a 
narrow  slit,  which  stands  parallel  to  the  edge  of  the  prism,  and 
a  lens  is  placed  in  the  path  of  the  light,  so  that  after  it  has 
passed  through  the  prism,  the  slit  is  sharply  focussed  on  the 
screen.  Each  different  color  which  exists  in  white  light  under- 
goes its  peculiar  refraction  in  the  prism,  and  the  spectrum 
obtained  is  really  a  row  of  differently  colored  images  of  the 
slit.  When  the  slit  is  very  narrow,  these  images  overlap  very 
little,  and  the  different  colors  are  obtained  with  as  little  ad- 
mixture of  others  as  possible.  Such  a  spectrum  is  called  a 
pure  spectrum. 

The  separation  of  white  light  into  its  constituent  colors  by 
refraction  is  called  dispersion.  The  length  of  the  spectrum 
obtained,  compared  with  the  deviation  of  the  whole  spectrum, 
is  called  the  dispersive  power  of  the  prism. 

139.  Chromatic  Aberration. — In  view  of  the  composite 
nature  of  white  light,  and  the  different  retfrangibilitiee  of  its 
constituent  colors,  it  is  evident  that  the  focus  of  a  point 


212  LIGHT 

source,  from  which  white  light  comes,  will  be  different  for  each 
of  the  constituent  colors,  and  hence  that  the  image  of  an  object 
viewed  through  a  lens  will  be  colored  around  the  edges.  Thia 
defect  of  the  image  is  said  to  be  due  to  chromatic  aberration. 
From  a  few  experiments  which  he  made,  Newton  concluded 
that  the  dispersive  power  of  a  prism  was  proportional  to  its 
refractive  power,  so  that  if  two  prisms,  or  two  lenses,  were 
superposed  in  such  a  way  as  to  annul  the  dispersion,  they 
would  at  the  same  time  annul  the  refraction.  Newton  there- 
fore decided  that  no  combination  of  lenses  could  be  made  which 
would  give  an  image  free  from  color.  He  accordingly  con- 
structed a  concave  spherical  mirror  to  be  used  instead  of  the 
object  glass  in  an  astronomical  telescope.  The  law  of  reflec- 
tion being  the  same  for  all  colors,  the  image  formed  by  this 
mirror  was  free  from  color.  Telescopes  of  this  sort  were  much 
superior  to  the  refracting  telescopes  then  in  use,  and  for  many 
years  quite  superseded  them. 

In  1757,  Dollond,  an  English  optician,  investigated  again 
the  relation  between  the  dispersive  and  refractive  powers  of 
different  kinds  of  glass,  and  found  that  no  such  proportion  be- 
tween them  prevailed  as  Newton  had  supposed.  This  being  so, 
it  became  possible  to  construct  a  combination  of  two  lenses, 
one  of  flint  glass  and  the  other  of  crown  glass,  so  shaped  as  to 
correct  the  dispersion  and  yet  capable  of  refracting  light  and 
forming  an  image.  Such  a  combination  is  called  an  achro- 
matic combination.  When  an  achromatic  combination  is  used 
as  the  object  glass  of  a  telescope  or  microscope,  the  image 
formed  by  it  is  almost  entirely  free  from  color  at  the  edges, 
and  by  the  use  of  another  achromatic  combination  as  eye 
piece,  the  disturbing  effects  of  dispersion  are  almost  entirely 
avoided. 

The  dispersive  power  of  different  materials  for  different 
colors  does  not  follow  any  general  law,  though  the  relative 
dispersive  powers  for  the  different  colors  are  nearly  the  same. 
The  achromatic  combination  of  two  lenses  can  therefore  only 
completely  correct  for  two  colors.  The  image  formed  by  it  is 
still  slightly  colored,  owing  to  this  so-called  irrationality  of 


LIGHT.  213 

dispersion.  By  adding  a  third  lens,  a  further  correction  can 
be  made.  In  the  finest  microscope  lenses,  made  after  the  de- 
signs of  Abbe,  the  corrections  for  both  spherical  and  chromatic 
aberration  have  been  pushed  so  far  that  the  lenses  are  practi- 
cally perfect. 

140.  Colors  of  Thin  Films.— The  attention  of  Newton  was 
directed  to  the  colors  seen  in  thin  films,  such  as  sheets  of  mica, 
or  the  walls  of  a  soap  bubble.  These  colors  had  been  already 
investigated  by  Hooke.  Newton's  most  important  observations 
were  made  by  the  help  of  the  film  of  air  between  a  plane  sheet 
of  glass,  and  a  slightly  convex  lens  laid  upon  it.  When  light 
was  allowed  to  fall  nearly  perpendicularly  upon  this  film,  and 
the  eye  was  placed  above  it,  a  succession  of  colored  rings  was 
seen,  having  as  a  common  centre  the  point  of  contact  between 
the  two  pieces  of  glass.  At  this  point  of  contact  there  was  a 
black  spot.  Newton  showed  that  when  light  of  one  color  was 
used,  the  rings  were  of  the  same  color,  and  that  the  diameter 
of  the  successive  rings  was  different  for  the  different  colors. 
The  diameter  of  the  ring  of  any  order  was  greatest  when  red 
light  was  used,  and  least  when  violet  light  was  used.  When 
light  of  one  color  was  used,  the  intervals  between  the  rings 
were  black.  From  this  observation  it  was  easy  to  explain  the 
colors  seen  with  white  light.  At  any  one  place  in  the  film  some 
of  the  constituents  of  the  white  light  will  not  appear,  or  will 
be  extinguished,  and  the  color  actually  seen  is  due  to  the  mix- 
ture of  the  remaining  constituents,  which  are  not  extinguished. 

From  the  known  curvature  of  the  lens  under  which  the 
film  was  formed,  Newton  calculated  the  thickness  of  the  film 
at  different  distances  from  the  black  spot  at  the  centre  of  the 
system  of  rings,  and  found  that  the  rings*  of  any  one  color 
appeared  at  parts  of  the  film,  whose  thicknesses  were  in  the 
proportion  of  the  numbers  1,  3,  5,  7,  etc.  This  result  indi- 
cated some  sort  of  periodicity,  or  regular  alternation  in  the 
properties  of  the  light. 

When  the  film  is  all  one  thickness,  the  color  seen  in  it  is 
the  same  in  all  parts,  except  for  slight  modifications  due  to 
the  different  angles  of  incidence  upon  the  film  of  the  light 
from  the  source. 


214  LIGHT. 

141.  Diffraction. — Other  systems  of  colored  bands  of  light 
were  first  observed  by  Grimaldi,  and  were  also  observed  by 
Newton.     They  occur  when  light,  which  has  passed  through 
a   very   small   opening,   passes   the   edge   of   an   obstacle,    or 
through  another  small  opening.    They  are  best  observed  if  the 
first  opening  is  a  narrow  slit.     In  this  case,  if  the  straight 
edge  of  a  screen  is  placed  in  the  light  coming  through  the  slit 
and  parallel  with  it,  the  shadow  of  this  edge  cast  on  a  re- 
ceiving screen  is  bordered  with  three  or  four  narrow  colored 
bands,  standing  outside  the  shadow.    The  edge  of  the  shadow 
itself  is  not  sharp,  as  it  should  be  if  light  traveled  in  geo- 
metrically straight  lines.     The  line  on  the  screen  in  which  it 
would  be  met  by  a  plane  passing  through  the  slit  and  the  edge 
of  the  obstacle  is  called  the  edge  of  the  geometrical  shadow. 
In  the  conditions  described,  the  colored  bands  stand  outside 
the  edge  of  the  geometrical  shadow,  and  the  illumination  ex- 
tends within  it,  gradually  fading  out  until  it  becomes  imper- 
ceptible. 

If  a  narrow  slit  is  placed  in  the  path  of  the  light  and 
parallel  with  the  first  slit,  the  light  which  comes  through  it  is 
a  narrow  band  of  white  light,  bordered  on  both  sides  by  a 
succession  of  bands  of  variously  colored  light. 

The  phenomena  here  described  and  others  like  them,  are 
said  to  be  due  to  the  diffraction  of  light. 

142.  Double  Refraction. — The   peculiar   phenomena   which 
appear  when  a  ray  of  light  falls  upon  a  crystal  of  calcite  or 
Iceland  spar  were  described  in  1669,  by  Bartholinus.    The  light 
undergoes  a  double  refraction.     That  is,  it  is  broken  by  re- 
fraction into  two  rays,  which  have  different  directions  in  the 
crystal.     If  the  incident  ray  is  perpendicular  to  the  surface, 
one  of  the  refracted  rays  is  also  perpendicular  to  the  surface. 
As  the  incident  ray  is  differently  inclined  to  the  normal,  this 
refracted  ray  follows  the  ordinary  law  of  refraction.     It  is 
therefore  called  the  ordinary  ray.    The  other  ray,  whose  law 
of   refraction  .Bartholinus   could   not   discover,    is   called    the 
extraordinary  ray.     If  a  surface  is  cut  on  the  crystal  per- 
pendicular to  the  axis  of  the  crystal,  the  incident  ray  per- 


LIGHT.  215 

pendicular  to  that  surface  is  not  doubly  refracted.  The  axis 
thus  determined  is  called  the  optic  axis. 

Huygens  investigated  the  double  refraction  of  Iceland 
spar,  and  discovered  the  law  by  which  the  refraction  of  the 
extraordinary  ray  may  be  described.  As  the  description  of 
this  law  depends  upon  the  theory  of  light  which  Huygens 
used  in  obtaining  it,  we  shall  postpone  the  consideration  of  it 
until  we  discuss  that  theory. 

Huygens  discovered  that  other  crystals  possess  the  prop- 
erty of  doubly  refracting  light,  and  subsequent  investigations 
have  shown  that  all  crystals,  except  those  of  the  isometric 
system,  are  doubly  refracting. 

143.  Polarization  of  Light. — In  his  investigation  of  Iceland 
spar,  Huygens  discovered  another  remarkable  property  of 
light.  To  show  it,  we  first  allow  a  narrovr  beam  of  light  to 
fall  perpendicularly  upon  one  face  of  a  crystal  of  Iceland  spar. 
The  light  is  then  doubly  refracted,  and  emerges  from  the 
opposite  face  in  two  nearly  parallel  beams.  If  a  second  crystal 
of  spar  is  placed  in  the  path  of  these  beams,  so  that  they  fall 
perpendicularly,  or  nearly  so,  upon  one  of  its  faces,  each  of 
them  will  ordinarily  be  doubly  refracted,  and  four  beams  of 
light  will  emerge  from  the  second  crystal.  A  difference  is 
observable  between  the  double  refraction  in  this  case,  and 
that  observed  in  the  first  crystal,  for  whereas  the  two 
emergent  beams  from  the  first  crystal  were  of  equal  intensity, 
the  two  beams  into  which  each  of  them  is  divided  by  the 
second  crystal  are,  in  general,  not  of  equal  intensity.  Indeed, 
by  turning  the  second  crystal  around  an  axis  parallel  with 
the  beam  of  light,  two  positions  of  it  may  be  found  in  which 
one  beam  of  each  pair  disappears  altogether,  while  the  other 
beam  of  each  pair  bus  the  intensity  of  the  beam  before  it  en- 
tered the  second  crystal.  We  may  describe  this  phenomenon 
otherwise  by  saying  that  the  first  crystal  divides  the  incident 
beam  into  an  ordinary  and  an  extraordinary  beam  of  equal 
intensities.  The  second  crystal  divides  each  of  these  into  an 
ordinary  and  an  extraordinary  beam,  whose  intensities  are 
generally  unequal.  One  position  of  the  second  crystal  can 


216  LIGHT. 

be  found  for  which  the  extraordinary  beam  of  the  first 
ordinary  beam,  and  the  ordinary  beam  of  the  first  extra- 
ordinary beam,  are  suppressed  or  disappear.  When  the 
crystal  is  turned  another  position  can  be  found,  in  which 
the  ordinary  beam  of  the  first  ordinary  beam,  and  the 
extraordinary  beam  of  the  first  extraordinary  beam,  are 
suppressed.  The  light  which  has  been  given  these  prop- 
erties, by  which  it  seems  to  differ  in  different  directions  in 
space  transverse  to  the  beam,  is  said  to  be  polarized. 

Huygens  could  give  no  explanation  of  polarization,  and  its 
exhibition  by  light  which  had  passed  through  Iceland  spar  was 
for  a  long  time  unique.  It  was  shown  much  later,  as  we  shall 
subsequently  consider  more  at  length,  that  whenever  double 
refraction  occurs,  the  two  rays  produced  are  always  polarized, 
and  it  was  discovered  also  that  light  may  be  polarized  by  ordi- 
nary reflection  and  refraction,  or  in  other  ways. 

144.  The  Emission  Theory  of  Light. — We  have  now  passed 
in  review  the  most  important  phenomena  exhibited  by  light 
which  were  known  to  the  earlier  workers  in  that  subject,  and 
which  had  to  be  considered  in  any  attempt  to  construct  a  theory 
of  light.  From  very  ancient  times  there  had  been  vaguely 
enunciated  two  rival  theories  of  light.  In  one  of  these,  light 
was  supposed  to  be  an  actual  emanation  of  some  substance 
from  the  luminous  body.  In  the  other,  it  was  assumed  that 
the  universe  is  pervaded  by  an  intangible  medium,  and  that 
light  consists  in  disturbances  transmitted  through  that 
medium.  In  the  form  which  this  latter  theory  took  at  the 
hands  of  Huygens,  these  disturbances  were  supposed  to  be 
waves,  similar  in  their  general  features  to  the  waves  of  sound 
in  air.  Huygens,  however,  could  not  adapt  this  theory  to  the 
explanation  of  the  transmission  of  light  in  straight  lines. 
When  the  waves  of  the  ocean  enter  the  narrow  opening  of  a 
harbor,  they  do  not  proceed  in  a  narrow  band  across  the  waters 
of  the  harbor,  leaving  the  water  on  either  side  of  the  band 
undisturbed,  but  they  spread  out  in  all  directions  from  the 
entrance.  In  the  same  way,  a  sound  which  is  made  inside  a 
building  will  spread  out  in  all  directions  when  it  passes 


LIGHT.  217 

through  an  open  window.  This  behavior  of  such  waves  as  we 
can  actually  experiment  with  seemed  to  Huygens'  critics  to 
make  the  wave  theory  of  light  untenable,  for  there  is  nothing 
more  conspicuous  about  the  behavior  of  light  than  its  transmis- 
sion in  straight  lines  and  the  fact  that  it  does  not  spread  out 
in  all  directions  after  it  has  passed  through  an  opening. 
Largely  on  this  account  Newton  refused  to  consider  the  wave 
theory,  and  set  himself  to  develop  its  rival,  the  emission  theory 
of  light.  A  brief  description  of  this  theory,  and  a  comparison 
of  it  with  the  wave  theory,  by  which  it  was  finally  displaced, 
will  furnish  an  excellent  illustration  of  the  methods  of  reason- 
ing used  in  constructing  a  physical  theory.  Newton  assumes 
that  a  luminous  body  shoots  out  from  itself  small  bodies,  which 
he  calls  corpuscles.  The  velocity  of  these  corpu-rlcs  is  that  of 
light.  Since  the  corpuscles  are  masses,  they  proceed  in 
straight  lines,  by  the  fundamental  law  of  inertia,  unless  they 
encounter  some  body  by  which  their  motion  is  changed.  By 
supposing  them  perfectly  elastic,  it  is  easy  to  show  that  when 
the  corpuscles  fall  obliquely  on  a  smooth  surface,  they  will 
rebound  from  it  in  a  direction  which  obeys  the  law  of  reflec- 
tion. By  supposing  further  that  the  corpuscles  are  sometimes 
strongly  attracted  by  ordinary  matter,  when  the  distance  be- 
tween the  corpuscle  and  the  matter  is  exceedingly  small,  and 
that  the  corpuscles,  having  entered  the  matter,  can  proceed 
through  it  without  interruption,  Newton  shows  that  the  path 
of  the  corpuscle  will  be  changed  in  direction  on  its  entering 
the  mass  of  matter  in  accordance  with  the  law  of  refraction. 
As  a  necessary  consequence  of  this  assumption,  the  velocity 
of  a  corpuscle  is  greater  in  the  more  highly  refracting  medium. 
To  account  for  the  different  colors  in  white  light,  Newton 
assumes  that  corpuscles  of  different  mass  and  perhaps  of  other 
different  properties  correspond  to  them.  The  colors  of  thin 
films  render  it  necessary  to  make  some  additional  hypothesis 
which  will  impart  to  the  corpuscles  some  sort  of  periodicity. 
The  particular  form  of  this  hypothesis  which  Newton  makes 
assumes  the  existence  of  an  elastic  medium,  called  by  Newton 
the  ether,  in  which  waves  are  set  up  whenever  a  corpuscle 


218  LIGHT. 

strikes  the  surface  of  a  body.  These  waves  travel  along  with 
the  corpuscle,  if  it  is  reflected,  and  with  nearly  the  same 
velocity,  so  that  when  it  meets  another  surface  the  wave  can 
be  thought  of  as  either  pushing  it  forward  to  help  it  pass 
through  the  surface,  or  as  pulling  it  backward  to  prevent  its 
passing  through.  Newton  names  these  two  conditions  the 
fits  of  easy  transmission  and  of  easy  reflection.  By  the  help 
of  these  fits  the  light  in  the  thin  film  acquires  the  necessary 
periodicity.  Another  and  an  easier  form  of  the  same  hypothe- 
sis is  that  given  by  Boscovich,  who  assumes  that  the  corpus- 
cles are  in  continual  rotation  around  axes  perpendicular  to 
their  paths,  and  that  their  ends  are  different,  so  that  as  the 
corpuscle  reaches  a  surface  it  will  be  reflected  if  one  of  its 
ends  is  presented  to  the  surface,  and  will  be  transmitted  if 
the  other  end  is  presented  to  the  surface.  In  order  to  explain 
the  bands  produced  by  diffraction,  Newton  makes  the  further 
supposition  that  the  corpuscles  which  pass  near  a  solid  edge 
move  back  and  forward,  as  he  describes  it,  with  a  motion  like 
that  of  an  eel,  and  so  proceed  in  one  direction  or  another 
according  to  the  direction  in  which  they  are  moving  when  they 
escape  the  influence  of  the  edge.  Newton's  ingenuity  fail* 
him  when  he  attempts  to  apply  this  theory  to  explain  double 
refraction,  and  he  cannot  explain  polarization  further  than 
by  saying  that  it  shows  that  light  has  sides.  In  the  subse- 
quent development  of  the  emission  theory,  which  held  its 
ground  for  a  century  and  a  half,  additional  hypotheses  were 
made  to  account  for  the  other  phenomena  which  were  dis- 
covered. It  cannot  escape  notice  that  this  theory,  which  at 
the  outset  is  so  simple,  and  which  accounts  so  well  for  recti- 
linear transmission,  and  for  reflection  and  refraction,  requires 
a  new  hypothesis  for  each  additional  phenomenon  which  is  to 
be  explained.  Each  hypothesis  is  of  service  only  in  explaining 
the  phenomenon  which  leads  to  its  adoption,  and  when  a  new 
phenomenon  is  considered,  a  new  hypothesis  has  to  be  made  to 
explain  it.  When  the  consequences  of  any  of  these  hypotheses 
are  followed  out,  no  new  and  previously  unknown  truths 
about  light  are  discovered.  The  only  necessary  conclusion 


LIGHT.  219 

from  the  fundamental  hypotheses  of  the  theory  which  was 
an  addition  to  the  knowledge  of  light,  turned  out  to  be  false 
when  it  was  examined  by  experiment.  As  has  been  stated 
already,  Newton's  principles  led  to  the  conclusion  that  light 
travels  faster  in  optically  denser  bodies.  This  conclusion  was 
tested  experimentally  by  Foucault,  in  1850,  and  found  to  be 
erroneous.  Light  really  travels  slower  in  optically  denser 
bodies. 

145.  The  Wave  Theory  of  Light. — The  wave  theory  of  light 
received  its  first  impulse  from  Huygens,  who  laid  down  the 
fundamental  principle  upon  which  its  development  depends. 
Huygens  perceived  that  if  a  train  or  succession  of  waves  is 
passing  through  a  medium,  the  disturbance  at  any  point  in 
the  medium  does  not  depend  solely  on  a  disturbance  coining 
directly  to  it  from  the  source,  but  is  the  resultant  of  all  the 
disturbances  which  reach  it  at  the  same  time  from  all  the 
disturbed  parts  of  the  medium.  To  put  it  otherwise,  if  a  point 
in  the  medium  is  disturbed  by  a  wave,  it  thereby  becomes  the 
centre  of  a  wave  disturbance,  which  goes  out  from  it  nearly 
as  if  it  were  a  centre  of  an  original  disturbance.  If  we  then 
consider  a  point  in  the  medium,  and  draw  a  spherical  surface 
around  it  as  centre,  and  suppose  each  of  the  points  on  that 
surface  to  be  in  some  way  disturbed  by  a  train  of  waves,  each 
of  those  points  will  send  a  wave  toward  the  centre  of  the 
sphere,  and  the  disturbance  at  the  centre  will  be  the  resultant 
effect  produced  by  the  waves  which  reach  it,  at  the  same 
instant,  from  the  different  points  of  the  surface.  In  order  to 
meet  the  facts  of  the  case,  especially  in  order  to  explain  that 
a  wave  disturbance  is  not  transmitted  backward  as  well  as 
forward,  Huygens  had  to  assume  that  the  elementary  wave 
sent  out  from  a  disturbed  point  does  not  transmit  equal  dis- 
turbances in  all  directions,  but  that  the  greatest  disturbance 
is  transmitted  along  the  ray,  in  the  direction  in  which  the 
wave  is  advancing  which  disturbs  the  point,  and  that  the  dis- 
turbances transmitted  in  oblique  directions  diminish  in  in- 
tensity until,  in  the  opposite  direction  to  that  of  the  ray,  they 
fall  off  to  nothing.  The  analysis  by  Kirchhoff  and  others  of 


220 


the  mode  of  transmission  of  waves  in  a  medium  shows  that 
this  hypothesis  of  Huygens  is  a  sound  one.  The  principle 
which  has  been  stated  is  known  as  Huygens'  principle. 

As  a  fundamental  hypothesis  of  the  wave  theory,  Huygens 
assumed  the  existence  of  an  ether,  or  universal  medium,  in 
which  the  waves  which  constitute  light  are  set  up  and  trans- 
mitted. This  ether  is  not  ordinary  matter.  It  is  not  cogniz- 
able by  the  senses.  It  occupies  space  that  is  otherwise  void, 
as  well  as  space  filled  with  ordinary  matter.  It  seems  to  be 
frictionless,  at  least  to  such  a  degree  that  slowly  moving 
bodies,  like  the  planets,  pass  through  it  without  loss  of  energy. 
It  possesses  certain  properties,  analogous  to  the  elasticity 
and  the  density  of  matter,  which  are  modified,  but  not  de- 
stroyed, by  the  presence  of  matter.  As  we  shall  see  later, 
these  properties  are  best  described  as  electric  properties,  but 
we  may,  for  the  present,  use  the  material  analogy.  Huygens 
supposed  that  waves  can  be  set  up  and  transmitted  in  this 
ether  according  to  the  principle  already  described,  and  then 
proceeded  to  give  explanations  of  certain  well-known  facts. 
These  explanations  we  shall  now  consider. 

Huygens  explained  the  rectilinear  transmission  of  light 
through  an  opening,  by  drawing  the  elementary  waves  around 
the  points  which  stand  in  the  line  of  the  opening,  and  showing 
that  directly  beyond  the  opening  these  waves  have  a  common 
envelope,  while  on  either  side  of  the  opening  they  have  not. 
He  supposed  the  only  effective  part  of  the  resulting  dis- 
turbance to  be  that  caused  by  this  common  envelope.  This 
explanation,  though  ingenious,  is  not  sufficient,  and  rectilinear 
transmission  was  not  really  explained  by  Huygens'  form  of 
the  wave  theory. 

To  give  Huygens'  explanation  of  reflection  and  refraction, 
we  make  the  following  construction.  We  draw  the  straight 
line  GH  to  represent  the  intersection  of  the  plane  of  the  paper 
with  a  plane  reflecting  surface  at  right  angles  to  it.  We  con- 
sider a  series  of  waves  advancing  obliquely  toward  this  sur- 
face in  air,  forming  a  beam,  which  is  bounded  by  the  extreme 
rays  EA  and  FC.  At  a  certain  instant,  one  of  the  wave  fronts 


LIGHT.  £21 

represented  by  the  line  AB  will  meet  the  surface  at  the  point 
A.  By  Huygens'  principle  that  point  will  become  the  centre 
of  a  wave  disturbance,  which  proceeds  from  it  in  both  media. 
We  shall  consider  it  first  in  the  second  medium.  At  the  end 
of  a  time  in  which  the  disturbance  which  was  at  B  has  reached 
the  point  C,  the  wave  from  A  has  become  a  sphere  with  A  as 
its  centre.  We  shall  suppose  that  the  velocity  of  light  in  the 
second  medium  is  less  than  that  in  the  first,  so  that  the  radius 
of  this  sphere  is  less  than  the  line  BC.  If  we  consider  the  light 
coming  from  any  point  between  the  points  A  and  B,  we  see 
that  it  will  reach  the  surface  GH,  at  some  point  between  A 
and  C,  after  part  of  the  above  mentioned  time  has  elapsed, 
and  that  there  will  proceed  out  from  that  point  a  spherical 
wave.  WThen  all  such  spherical  waves  are  constructed,  it  will 
be  found  that  they  have  a  common  tangent,  which  is  a  line 
drawn  from  the  point  C.  This  line  is  therefore  the  envelope 
of  all  the  elementary  waves  which  correspond  to  the  original 
disturbances  in  the  wave  front  AB,  and  therefore  represents 
the  refracted  wave.  If  we  draw  from  the  point  A  the  radius 
AD  to  the  point  of  tangency,  the  direction  thus  determined  is 
the  direction  of  the  refracted  beam.  The  lines  BC  and  AD  are 
distances  passed  over  by  the  reflected  and  refracted  waves 
respectively  in  the  same  time,  and  they  are  therefore  propor- 
tional to  the  velocities  of  light  in  the  first  and  second  media. 
From  the  construction  we  see  that  these  same  lines  are  pro- 
portional to  the  sines  of  the  angles  of  incidence  and  refraction. 
The  ratio  of  these  sines  is  therefore  equal  to  the  ratio  of  the 
velocities  of  light  in  the  two  media,  and  this  ratio  is  pre- 
sumably constant  for  the  two  media.  We  therefore  deduce  in 
this  way  the  law  of  refraction.  Huygens'  hypothesis  that  the 
velocity  of  light  is  less  in  the  denser  medium,  which  is  a 
necessary  hypothesis  of  the  wave  theory,  was  confirmed  by 
the  experiments  of  Foucault  already  referred  to. 

A  construction  that  is  essentially  similar,  in  which  the 
elementary  waves  from  the  points  on  the  lines  A  C  are  drawn 
in  the  upper  medium,  will  lead  to  the  law  of  reflection. 

By  the  use  of  the  wave  theory,  Huygens  was  able  to  give 


222  LIGHT. 

a  description  of  the  double  refraction  in  Iceland  spar.  To  do 
so  he  made  the  supposition  that  the  disturbance  which  reaches 
a  point  on  the  surface  of  the  spar  sets  up  two  waves,  one  of 
which  is  spherical,  and  corresponds  to  the  ordinary  ray.  The 
other  is  an  ellipsoid  of  revolution,  having  its  axis  of  revolu- 
tion parallel  to  the  axis  of  the  crystal.  By  a  construction 
which  is  essentially  similar  to  the  one  we  have  used  in  ex- 
plaining ordinary  refraction,  Huygens  was  able  to  determine 
the  direction  of  the  extraordinary  ray  for  various  incidences, 
and  to  show  that,  in  each  case,  it  was  in  complete  accord  with 
his  observations. 

Huygens  did  not  develop  the  wave  theory  beyond  this 
point.  In  the  form  in  which  he  gave  it,  it  was  really  defective 
in  that  it  did  not  explain  satisfactorily  rectilinear  trans- 
mission. The  emission  theory,  therefore,  svipported  by  the 
great  authority  of  Newton,  remained  prevalent  until  another 
set  of  phenomena  was  discovered,  with  which  the  development 
of  the  wave  theory  really  begins. 

146.  Interference  of  Light. — In  1803  Young  discovered  an 
effect  produced  by  light,  which  of  itself,  and  because  of  the 
consequences  deduced  from  it,  was  almost  conclusive  in  favor 
of  the  wave  theory. 

Young  allowed  a  beam  of  light  to  pass  through  a  very 
small  opening,  and  received  it  on  a  distant  screen.  On  inter- 
posing a  narrow  strip  of  card  in  the  path  of  the  beam,  and 
observing  its  shadow  on  the  screen,  he  perceived  that  it  was 
not  uniform,  but  contained  a  number  of  alternately  light  and 
dark  bands,  parallel  with  its  edges.  He  showed  that  these 
bands  were  due  to  the  combination  of  the  two  portions  of  the 
beam  which  had  passed  the  two  edges  of  the  card;  for,  on 
cutting  off  the  light  on  one  side  of  the  card  with  another 
screen,  the  bands  disappeared. 

Young  subsequently  described  another  form  of  this  experi- 
ment, which  presents  its  essential  features  very  clearly.  In 
it,  light  which  has  passed  through  a  very  small  opening  is 
allowed  to  fall  on  two  small  holes 'or  parallel  slits  set  close 
together.  On  observing  the  light  cast  through  these  holes  on 


LIGHT.  223 

a  screen  behind  them,  the  region  in  which  the  two  beams  of 
light  overlap  is  seen  to  be  crossed  by  a  set  of  parallel  light 
and  dark  bands.  When  either  of  the  holes  through  which  the 
light  comes  is  covered,  these  bands  disappear.  They  cannot 
be  obtained  when  the  light  which  falls  on  the  openings  comes 
directly  from  an  ordinary  source.  This  experiment  is  known 
as  Young's  experiment. 

Young  explained  these  phenomena  by  the  help  of  the  wave 
theory.  To  explain  the  second  experiment,  we  suppose  that 
the  waves,  starting  from  the  first  opening,  which  we  shall 
hereafter  call  the  source,  set  up  waves  at  each  of  the  two 
openings  in  the  screen,  these  waves  proceeding  from  the  open- 
ings almost  as  if  they  were  independent  sources.  If  we  con- 
sider the  effect  at  any  one  point  on  the  receiving  screen  pro- 
duced by  both  these  waves,  we  see  by  construction  that  it 
will  depend  on  the  relative  distances  of  the  point  from  the 
two  openings.  When  the  difference  of  these  distances  is  equal 
to  half  a  wave  length,  or  any  odd  multiple  of  half  a  wave 
length,  the  two  waves  will  reach  the  point  in  opposite  phases, 
and  by  the  principle  of  the  superposition  of  small  motions, 
the  resultant  effect  will  be  zero.  The  black  bands  that  are 
seen  correspond  to  this  condition.  At  intermediate  points 
between  the  black  bands,  for  which  the  difference  of  the  dis- 
tances to  the  openings  is  an  even  number  of  half  wave  lengths, 
the  waves  from  the  openings  will  meet  in  similar  phases.  They 
will  therefore  increase  the  light  received  at  those  points. 

We  explain  the  first  experiment  in  an  essentially  similar 
way,  considering  that  the  two  beams  of  light  which  combine 
to  produce  the  light  and  dark  bands  are  those  portions  of  the 
original  beam  which  pass  close  to  the  edges  of  the  card  and 
are  turned  into  its  shadow  by  diffraction. 

The  most  striking  single  argument  that  was  ever  adduced 
in  favor  of  the  wave  theory  was  that  presented  by  the  disap- 
pearance of  the  system  of  bands  when  the  light  from  one  of 
the  openings  was  intercepted,  and  their  reappearance  when  it 
was  again  allowed  to  pass.  This  effect  follows  as  a  natural 
consequence  of  the  wave  theory  and  is  inconsistent  with  any 


224  LIGHT. 

reasonable  form  of  the  emission  theory,  for  it  seems  impossible 
to  explain  how  the  meeting  of  two  streams  of  particles  will 
produce  an  effect  different  from  that  which  would  be  produced 
by  either  stream  of  particles  acting  alone. 

Waves  which  are  superposed  on  each  other,  in  the  way 
which  has  been  "described,  so  as  to  produce  variations  of 
intensity  in  different  parts  of  the  field  through  which  they 
are  passing,  are  said  to  interfere,  and  the  general  phenomenon 
is  called  the  interference  of  light.  The  waves  do  not  interfere 
with  each  other  in  such  a  way  as  to  destroy  each  other's  in- 
dividuality. Each  wave  passes  through  any  point,  and  pro- 
ceeds beyond  it,  as  if  no  other  wave  had  met  it  at  that  point. 
This  follows  as  a  consequence  of  the  principle  of  superposition. 
All  that  is  meant  by  interference  is  that  the  waves  combine 
or  interfere  with  each  other  at  any  point  as  respects  their 
individual  effects,  so  that  the  effect  perceived  is  that  due  to 
their  combined  action. 

As  soon  as  Young  had  discovered  interference,  he  found  that 
with  its  help  he  could  apply  the  wave  theory  to  explain  the 
colors  of  thin  films.  Consider  a  train  of  waves  incident  on  the 
upper  surface  of  a  film.  A  normal  to  these  waves  will  meet 
that  surface  at  a  point  A,  and  another  normal,  at  the  neighbor- 
ing point  B.  The  waves  which  reach  A  are  partly  reflected  and 
partly  refracted  into  the  film.  The  wave  in  the  film  proceeds 
in  the  direction  of  its  normal  to  the  point  C  on  the  second  face 
of  the  film,  where  it  will  again  undergo  reflection  within  the 
film,  and  refraction  into  the  outside  medium.  The  reflected 
part  will  proceed  again  through  the  film,  and  part  of  it  will 
emerge  at  the  point  B.  At  that  point  it  will  be  superposed  on 
a  wave  directly  reflected  there,  and  the  effect  produced  in  the 
direction  of  the  reflected  wave  normal  will  be  due  to  the  com- 
bination of  these  waves.  If  they  are  in  the  same  phase  they 
will  produce  an  enhanced  effect,  so  that  light  of  the  peculiar 
color  which  corresponds  to  the  wave  length  will  be  seen.  If 
they  are  in  opposite  phases,  they  will  annul  each  other  and  no 
light  will  be  seen.  When  white  light  is  used,  which  we  may 
suppose  to  contain  waves  of  all  periods  lying  between  certain 


LIGHT.  225 

limits,  this  description  shows  that  some  of  its  constituents  will 
be  suppressed,  while  others  will  not  be,  so  that  the  reflected 
light  will  appear  colored,  and  the  color  will  differ  for  different 
thicknesses  of  the  film. 

When  we  consider  the  relation  between  the  thickness  of  the 
film  and  the  wave  length  of  the  light  which  is  enhanced  or 
suppressed,  it  seems  at  first  sight  that  the  two  waves  from  the 
point  B  would  be  in  similar  phases  when  the  distance  AC+CB, 
or  twice  the  thickness  of  the  film,  is  equal  to  the  wave  length, 
or  to  any  multiple  of  the  wave  length  of  light  in  the  film.  This 
turns  out  to  be  not  the  case.  In  the  conditions  described  no 
light  leaves  the  point  B.  To  account  for  this  Young  assumed 
that  the  reflection  of  light  takes  place,  in  different  circum- 
stances, with  or  without  change  of  sign.  At  one  of  the  sur- 
faces, say  at  the  upper  one,  the  light  is  passing  from  a  denser 
to  a  rarer  medium,  and  from  the  analogy  of  the  reflection  of 
sound,  we  may  suppose  it  to  be  reflected  there  without  change 
of  sign.  At  the  other  surface  it  passes  from  a  rarer  to  a 
denser  medium,  and  we  accordingly  suppose  it  to  be  reflected 
there  with  change  of  sign.  In  this  way  the  light  which  is  re- 
flected in  the  film  has  its  phase  reversed,  and  in  considering  its 
combination  with  the  light  reflected  on  the  upper  surface,  we 
must  consider  its  phase  opposite  to  that  which  would  be  due 
to  the  distance  over  which  it  travels  in  the  film.  With  this 
additional  hypothesis,  which  subsequent  analysis  has  shown  to 
be  justified,  Young  was  able  to  explain  the  colors  of  thin  films, 
and  the  size  of  the  successive  rings  in  Newton's  experiment, 
and  to  show  that  the  wave  lengths  of  the  different  colors,  cal- 
culated from  the  size  of  the  rings,  were  in  agreement  with  those 
obtained  by  him  from  his  original  experiments  on  interference. 

If  we  trace  the  course  of  the  waves  which  emerge  after 
passing  through  the  film,  we  find  that,  of  the  waves  which  are 
superposed  at  the  point  D,  one  of  them  has  passed  through  the 
film  once,  while  the  other  has  passed  through  it  three  times, 
and  has  besides  undergone  two  reflections  with  change  of  sign. 
The  difference  of  phase  of  the  waves  superposed  at  D  will  there- 
fore depend  only  on  the  difference  of  their  paths  in  the  film, 


226  LIGHT. 

and  light  will  emerge  when  the  difference  in  path  is  equal  to 
an  even  number  of  half  wave  lengths  in  the  film.  Light  is  thus 
transmitted  through  the  film  when  its  thickness  is  such  that 
no  light  is  reflected  by  it.  In  Newton's  experiment  a  system 
of  rings  will  be  seen  by  transmitted  light  which  is  complemen- 
tary to  those  seen  by  reflected  light;  that  is,  in  which  the 
colors  are  so  arranged  that  when  they  are  superposed  on  the 
reflected  system,  a  uniform  field  of  white  light  is  the  result. 

A  more  extended  study  of  the  colors  of  thin  films,  in  which 
account  is  taken  of  the  fact  that  the  light  which  enters  the 
film  does  not  all  leave  it  after  one  or  two  internal  reflections, 
confirms  the  conclusions  of  the  elementary  theory  in  the  case 
of  the  reflected  light,  but  shows  that  the  transmitted  light  will 
never  be  entirely  extinguished  for  any  thickness  of  the  film. 
As  the  thickness  of  the  film  changes,  the  transmitted  light  will 
pass  through  a  series  of  maximum  and  minimum  values.  These 
conclusions  are  confirmed  by  observation. 

Young  also  applied  the  wave  theory  to  explain  diffraction. 
He  supposed  that  the  diffraction  bands  outside  the  shadow  are 
caused  by  the  interference  of  the  light  which  passes  the  obstacle 
outside  its  geometrical  shadow,  with  light  reflected  from  the 
edge  of  the  obstacle.  In  this  way  he  could  explain  in  general 
the  production  of  the  diffraction  bands,  but  he  could  not  ac- 
count for  their  exact  position,  nor  could  he  readily  explain 
some  other  of  the  diffraction  phenomena. 

147.  Fresnel's  Development  of  the  Wave  Theory. — A  few 
years  after  Young's  discovery  of  interference,  Fresnel  began 
researches  by  which  the  wave  theory  of  light  was  carried  to  a 
very  high  development.  His  early  experiments  on  interference 
were  similar  to  Young's,  but  he  made  an  important  step  in 
advance  by  demonstrating  the  interference  of  light  in  a  way 
that  met  an  objection  which  had  been  raised  to  Young's  experi- 
ment. In  regard  to  that  experiment  it  was  said  that  the  black 
bands  observed  by  Young  were  not  produced  by  simple  inter- 
ference, but  were  diffraction  phenomena  due  to  an  action  upon 
the  light  of  the  edges  of  the  opening  through  which  it  passed. 
Fresnel  avoided  this  objection  by  substituting  for  the  two  open- 


LIGHT.  227 

ings  two  images  of  the  source  formed  in  two  plane  mirrors, 
which  met  along  one  edge,  and  were  very  slightly  inclined  to 
each  other.  By  properly  adjusting  the  inclination  of  these 
mirrors,  the  two  beams  of  light  reflected  from  them  were  made 
to  overlap,  and  interference  bands,  similar  to  those  observed  by 
Young,  were  produced.  Fresnel  found  that  by  placing  an  eye 
piece  in  the  path  of  these  beams,  the  interference  bands  could 
be  observed  in  it.  He  was  thus  able  to  observe  them  and  to 
measure  their  distances  much  more  precisely  than  can  be  done 
when  they  are  allowed  to  fall  on  a  screen.  All  the  interfer- 
ence and  diffraction  phenomena  can  be  observed  best  in  this 
way. 

It  was  evident  that  the  wave  theory,  however  successful  it 
might  be  in  explaining  some  of  the  phenomena  of  light,  could 
not  be  accepted  if  it  could  not  be  made  to  account  for  the  recti- 
linear transmission  of  light.  By  a  combination  of  Huygens' 
principle  with  Young's  principle  of  interference,  Fresnel  was 
not  only  able  to  account  for  rectilinear  transmission,  but  also 
to  explain  by  the  same  principles  all  the  phenomena  of  diffrac- 
tion. To  describe  Fresnel's  explanation  of  rectilinear  trans- 
mission of  a  linear  wave,  we  construct  a  diagram  as  follows: 
Supposing  the  source  of  light  to  be  a  point  at  an  infinite  dis- 
tance on  the  left,  we  draw  a  straight  line  to  represent  one  of 
the  rays  sent  from  it,  and  at  right  angles  to  this  line  we  draw 
a  number  of  equidistant  straight  lines.  The  distance  between 
two  of  these  lines  is  taken  to  be  half  a  wave  length.  The  suc- 
cessive lines,  therefore,  represent  parts  of  the  successive  waves 
which  are  in  opposite  phases.  Take  a  point  on  the  ray  and  with 
it  as  centre  describe  a  circle.  By  the  principle  of  Huygens 
every  point  on  this  circle,  being  disturbed  by  the  waves  passing 
over  it,  will  act  as  a  centre  of  disturbance  and  will  send  out  a 
wave.  The  waves  from  all  the  points  on  the  circle  will  reach 
the  centre  at  the  same  instant,  and  their  phases  at  that  centre 
will  differ  in  the  same  way  that  they  differ  in  the  circle.  The 
effect  at  the  centre  will  be  that  due  to  their  superposition. 
Now,  if  we  examine  those  portions,  or  elements,  of  the  circle 
which  are  intercepted  between  the  parallel  lines  of  the  figure, 


we  see  that  the  disturbances  from  two  successive  elements  will 
in  general  be  in  opposite  phases,  so  that  when  they  are  super- 
posed at  the  centre,  they  tend  to  destroy  each  other.  If  we  at 
first  confine  our  attention  to  that  portion  of  the  circle  which 
lies  nearer  the  source,  we  see  at  once  that  much  the  largest 
element  is  that  which  lies  nearest  the  line  joining  the  source 
with  the  centre  of  the  circle,  and  that  the  successive  elements, 
as  we  go  out  around  the  circle,  soon  become  appreciably  equal 
to  each  other.  It  is  natural  to  suppose  that  the  effect  of  any 
one  of  these  elements  is  proportional  to  its  length.  Those  ele- 
ments which  are  nearly  equal  to  each  other,  therefore,  annul 
each  other's  effects  in  pairs,  and  the  effect  at  the  centre  is  due 
to  the  very  considerable  preponderance  of  the  effect  of  the  first 
element  over  that  of  those  which  lie  outside  of  it. 

If  we  consider  that  portion  of  the  circle  which  lies  farther 
from  the  source  than  the  centre,  we  shall  find  the  same  differ- 
ence in  the  size  of  the  successive  elements  as  we  found  before. 
We  may,  however,  neglect  the  action  of  this  part  of  the  circle 
if  we  remember  that  in  the  application  of  Huygens'  principle 
we  have  agreed  to  suppose  that  the  effect  produced  by  the  ele- 
mentary waves  diminishes  with  the  obliquity,  and  that  in  par- 
ticular their  effect  vanishes  in  the  direction  opposite  to  that 
in  which  the  actual  waves  progress. 

'  It  thus  appears  thdt  the  effect  which  reaches  the  centre  of 
the  circle  is  practically  due  entirely  to  that  element  of  the 
circle  which  lies  nearest  the  straight  line  joining  the  source 
with  the  centre.  The  light  received  at  the  centre,  therefore, 
appears  to  travel  from  the  source  in  a  straight  line.  The  con- 
clusion which  we  have  drawn  from  this  .description  is  fully 
confirmed  by  analysis. 

Fresnel  applied  similar  principles  to  the  explanation  of  the 
diffraction  produced  by  the  edge  of  an  obstacle.  The  linear 
waves  which  cause  the  effect  are  perpendicular  to  the  edge  as 
they  pass  the  obstacle.  To  consider  this  case  we  construct  the 
diagram  as  before,  and  suppose,  first,  that  the  obstacle  is  so 
interposed  that  the  waves  which  would  pass  the  circle  are  all 
intercepted.  As  the  obstacle  is  lowered,  waves  are  set  up  at 


LIGHT.  229 

the  different  portions  of  the  circle  in  succession.  Those  from 
the  uppermost  parts  of  the  circle  have  no  effect,  on  account  of 
the  obliquity,  but  as  the  obstacle  is  still  further  lowered,  waves 
are  at  last  set  up  in  elements  which  send  appreciable  effects  to 
the  centre  of  the  circle.  As  we  have  already  seen,  the  element 
nearest  to  the  obstacle  will  be  the  largest,  and  it  will  send  an 
effect  to  the  centre  that  will  not  be  entirely  annulled  by  the 
effects  of  the  other  elements.  As  the  edge  of  the  obstacle  ap- 
proaches the  line  from  the  source  to  the  centre,  the  element 
nearest  to  it  becomes  relatively  larger  and  its  effect  relatively 
greater.  When  the  edge  of  the  obstacle  is  on  that  line,  the 
effect  at  the  centre  is  that  produced  by  half  the  wave.  The 
wave  theory,  therefore,  leads  naturally  to  an  explanation  of  the 
illumination  of  the  geometrical  shadow.  If  the  obstacle  is  still 
further  lowered,  so  as  to  expose  the  first  element  on  the  other 
side  of  the  line  joining  the  source  with  the  centre,  the  effect 
of  that  element  is  added  to  that  of  the  upper  half  of  the  circle 
and  the  centre  is  more  highly  illuminated.  If  the  obstacle 
exposes  two  elements  below  the  line,  their  effects  at  the  centre, 
being  in  opposite  phases,  partly  destroy  each  other  and  the 
centre  is  relatively  darker.  Similar  alternations  of  relative 
brightness  and  darkness  occur  at  the  centre  as  the  successive 
elements  are  exposed.  The  wave  theory  thus  accounts  for  the 
diffraction  bands  seen  outside  the  geometrical  shadow. 

When  Fresnel  made  a  calculation  of  the  positions  of  the 
diffraction  bands,  using  the  wave  length  of  light  which  he  had 
already  obtained  from  interference  experiments,  he  found  that 
the  calculated  positions  agreed  with  the  observed  positions  of 
the  bands. 

We  may  consider  also  the  diffraction  produced  by  a  narrow 
slit.  If  we  take  a  limited  line  whose  length  is  the  width  of  the 
slit  and  move  it  across  our  diagram,  the  elements  which  it  will 
cover  will  be  those  exposed  by  the  slit,  and  we  may  determine 
the  effect  produced  by  the  slit  by  considering  the  elements  of 
the  circle  which  will  be  exposed.  If,  for  example,  the  slit  is  so 
placed  as  to  expose  the  two  central  elements,  the  centre  of  the 
circle  will  be  brightly  illuminated.  If,  on  the  other  hand,  it 


230  LIGHT. 

exposes  the  first  and  the  second  element,  they  will  partly  coun- 
teract each  other  and  the  centre  will  be  relatively  darker.  If 
it  exposes  three  elements,  the  third  will  supplement  the  effect 
of  the  first  and  the  centre  will  be  brighter  again.  In  general 
the  centre  will  be  illuminated  if  the  slit  exposes  an  odd  number 
of  elements  and  will  be  dark  if  it  exposes  an  even  number. 

On  the  natural  assumption  that  the  color  of  light  depends 
on  its  wave  length,  it  is  easy  to  explain  the  colors  in  the  dif- 
fraction bands.  For,  because  of  the  difference  of  wave  length, 
the  elements  of  our  diagram  will  differ  in  length,  for  the  differ- 
ent colors,  and  a  certain  position  of  the  slit  which  will  produce 
darkness  at  the  centre  for  one  wave  length  will  not  do  so  for 
another.  For  different  positions  of  the  slit,  therefore,  if  we 
use  white  light,  the  centre  will  be  illuminated  with  different 
colors.  Consideration  of  the  diagram  will  make  it  plain  that 
the  shorter  the  wave  length,  the  less  does  the  slit  have  to  be 
moved  out  from  the  central  line  in  order  to  expose  an  odd 
number  of  elements  and  so  to  cause  illumination  at  the  centre. 
Therefore,  when  white  light  falls  on  a  slit,  the  diffracted  light 
of  shortest  wave  length  lies  nearest  the  white  band  which 
passes  straight  through  the  slit.  The  deviation  of  the  colors 
is  the  reverse  of  that  caused  by  dispersion.  The  wave  length 
of  the  violet  is  the  least,  and  that  of  the  red,  the  greatest,  of 
the  colors  which  constitute  the  spectrum. 

It  may  be  as  well  to  mention  here,  though  it  was  not  used 
by  Fresnel,  the ,  instrument  called  the  diffraction  grating.  It 
consists  of  a  very  large  number  of  equidistant  slits.  These  are 
made  by  cutting  lines  with  a  diamond  upon  a  glass  plate  or 
upon  a  plate  of  speculum  metal.  It  may  be  shown  by  theory 
that  when  light  of  one  wave  length  falls  upon  such  a  grating 
from  a  point  source,  the  diffraction  bands  produced  are  very 
intense  and  narrow.  The  distance  between  them  depends  on 
the  distance  between  the  successive  cuts  on  the  grating,  and 
may  be  made  large  by  making  the  cuts  near  together.  In  the 
gratings  used  in  most  spectroscopic  work,  there  are  from  15,000 
to  20,000  cuts  to  the  inch,  and  the  diffraction  bands  are  so 
widely  separated  that  only  three  or  four  of  them  can  appear 


LIQBT.  231 

in  front  of  the  grating.  When  white  light  falls  on  such  a  grat- 
ing, its  constituent  colors  are  diffracted  at  different  angles, 
according  to  their  wave  lengths.  With  fine  gratings,  having  a 
large  number  of  lines,  the  spectrum  thus  obtained  is  of  great 
purity.  Since  the  distance  between  the  diffraction  bands  pro- 
duced by  different  wave  lengths  is  proportional  to  the  difference 
between  the  wave  lengths,  this  spectrum  is  called  a  normal 
spectrum. 

148.  Polarized  Light. — In  the  form  of  the  wave  theory  first 
used  by  Fresnel,  the  waves  were  thought  of  as  longitudinal. 
That  is,  the  vibrations  of  the  medium  were  supposed  to  take 
place  to  and  fro  in  the  line  of  transmission.  An  example  of 
such  a  mode  of  vibration  was  already  known  in  the  case  of 
sound,  and  it  was  also  known  that  a  fluid,  as  the  ether  was 
then  assumed  to  be,  could  only  vibrate  in  that  manner.  When 
Young  and  Fresnel  tried  to  apply  the  wave  theory  to  the  ex- 
planation of  double  refraction  and  polarized  light,  they  found 
it  impossible  to  make  any  headway  so  long  as  they  retained 
the  hypothesis  of  longitudinal  vibrations.  In  fact  it  is  clear 
that  such  vibrations  cannot  present  any  such  distinct  differ- 
ences on  different  sides  of  the  ray  as  occur  in  the  case  of  polar- 
ized light.  Both  Young  and  Fresnel  determined,  therefore,  to 
abandon  the  hypothesis  of  longitudinal  vibrations,  and  to  adopt 
in  its  stead  the  hypothesis  that  the  vibrations  are  more  or 
less  transverse  to  the  line  of  progress.  Young  did  little  more 
than  indicate  his  adoption  of  this  view,  and  its  development 
was  entirely  due  to  Fresnel. 

In  order  to  verify  the  hypothesis  of  transverse  vibrations, 
so  far  as  it  can  be  done  by  experiment,  Fresnel  and  his  friend 
Arago  executed  a  series  of  experiments  on  the  interference  of 
polarized  light.  To  understand  the  statement  of  their  results, 
it  must  be  mentioned  that  the  two  polarized  rays  which  pass 
through  a  crystal  of  Iceland  spar  are  said  to  be  polarized  in 
opposite  planes,  meaning  thereby  in  planes  at  right  angles  to 
each  other,  but  parallel  with  the  direction  of  the  ray.  Fresnel 
and  Arago  tried  Young's  experiment  to  obtain  interference, 
only  using  polarized  light,  variously  modified,  instead  of  nat- 


282  LIGHT. 

ural  light.  They  found  that  when  polarized  light  was  used  as 
the  source  and  was  not  modified  before  it  fell  upon  the  two 
openings,  the  interference  phenomena  obtained  were  just  the 
same  as  those  obtained  with  ordinary  light.  By  interposing 
properly  prepared  crystals  in  the  paths  of  the  polarized  light 
falling  from  the  source  on  the  two  openings,  they  polarized  the 
beams  which  fell  on  the  openings  in  opposite  planes,  and  then 
found  that  interference  did  not  occur.  This  result  is  in  accord 
with  the  hypothesis  that  the  vibrations  of  light  are  transverse 
to  the  line  of  progress,  and  are  at  right  angles  to  each  other  in 
oppositely  polarized  rays.  For,  when  the  rays  from  the  two 
openings  in  the  first  experiment  were  similarly  polarized,  they 
would  be  in  the  same  plane  and  so  could  interfere  destructively. 
When,  in  the  second  experiment,  the  two  rays  were  polarized 
in  opposite  planes,  the  vibrations  would  be  perpendicular  to 
each  other,  and  so  could  never  destroy  each  other  by  interfer- 
ence ;  for  it  is  plain  that  two  vibrations  at  right  angles  to  each 
other  can  never  act  on  one  particle  so  as  to  keep  it  at  rest. 
Fresnel  concluded  from  these  experiments  that  the  hypothesis 
of  transverse  vibrations  was  confirmed. 

It  may  be  well  to  consider  at  this  point  Fresnel's  descrip- 
tion of  common  light  on  the  hypothesis  of  transverse  vibra- 
tions. He  supposed  it  to  be  such  a  motion  in  the  medium  as 
may  be  obtained  by  the  superposition  of  two  simple  harmonic 
motions  transverse  to  the  line  of  progress  and  at  right  angles 
to  each  other.  The  path  of  a  point  describing  such  a  motion 
is,  in  general,  an  ellipse.  Two  such  motions  may  be  superposed 
so  as  to  produce  interference  so  long  as  the  elliptic  paths  are 
similar.  The  fact  that  interference  can  be  obtained  between 
two  rays  of  light  which  differ  in  length  by  2,500,000  wave 
lengths,  shows  that  the  vibrations  at  the  source  remain  sim- 
ilar, and  send  out  similar  disturbances  through  space,  for  a 
time  containing  at  least  2,500,000  periods  of  the  vibration.  On 
the  other  hand,  it  is  also  evident  that  the  vibration  from  a 
source  does  not  remain  always  the  same.  If  it  did  do  so,  it 
would  be  polarized,  and  the  two  rays  into  which  it  is  broken 
by  a  doubly  refracting  crystal  would  differ  in  intensity  from 


LIGHT.  233 

each  other,  and  would  have  different  relative  intensities  for 
different  positions  of  the  crystal.  Now  observation  shows  that 
both  the  rays  transmitted  by  the  crystal  have  the  same  inten- 
sity, when  the  light  used  is  common  light.  We  can  explain 
this  only  by  supposing  that  the  phase  of  one  of  the  component 
vibrations,  or  the  phases  of  both  of  themj  change  abruptly  from 
time  to  time,  so  as  to  alter  the  polarization  of  the  vibration. 
Since  over  500  million  million  vibrations  are  executed  by  yel- 
low light  in  one  second,  a  sufficient  number  of  such  changes 
may  occur  in  that  time  to  give  the  two  component  vibrations 
into  which  the  common  light  is  divided  by  the  crystal  equal 
average  intensities,  and  yet  those  vibrations  may  continue  in 
one  phase  long  enough  to  account  for  the  interference  of  rays 
whose  paths  differ  by  millions  of  wave  lengths.  This  descrip- 
tion of  common  light  is  confirmed  by  another  experiment  of 
Fresnel  on  the  interference  of  polarized  light.  He  found  that 
if  light  originally  polarized  before  it  entered  the  source  was 
divided  into  two  beams  polarized  at  right  angles,  which  fell  on 
the  two  openings,  these  beams,  which,  in  this  condition,  would 
not  interfere,  could  be  made  to  interfere,  if  their  planes  of 
polarization  were  brought  into  coincidence.  On  the  contrary, 
if  the  light  which  fell  on  the  source  was  not  polarized,  and  if 
its  two  beams  which  fell  on  the  openings  were  first  polarized 
in  opposite  planes,  the  beams  thus  formed  did  not  interfere, 
even  when  they  were  brought  into  the  same  plane  of  polariza-  • 
tion.  In  the  latter  experiment,  we  notice  that  the  two  polar- 
ized beams  which  fall  on  the  two  openings  contain  the  two 
components  of  the  elliptic  vibration  of  common  light.  Accord- 
ing to  our  hypothesis,  these  components  change  their  phases 
occasionally,  and  there  is  no  reason  to  think  that  their  changes 
of  phase  will  always  occur  together.  If,  therefore,  those  com- 
ponents of  them  which  fall  in  the  similarly  polarized  beams 
upon  the  receiving  screen  are  at  one  instant  so  related  as  to 
interfere  at  one  point,  they  will  not  remain  so  long  enough  for 
the  interference  to  be  perceptible.  The  interference  bands  will 
pass  from  one  point  to  another  on  the  screen  so  many  times 
in  a  second  that  the  illumination  of  the  screen  will  appear 
uniform. 


234  LIGHT. 

Fresnel  employed  the  hypothesis  of  transverse  vibrations  to 
explain  polarization  by  reflection.  It  had  been  discovered  a  few 
years  before  by  Malus  that  when  light  is  incident  upon  a  re- 
flecting surface  of  water  or  glass,  the  light  in  both  the  reflected 
and  refracted  beams  is  generally  partially  polarized.  For  a 
certain  angle  of  incidence,  called  the  polarizing  angle,  the  re- 
flected light  is  completely  polarized.  Most  reflecting  sub- 
stances, except  the  metals,  have  this  property  of  polarizing 
light  by  reflection.  It  was  discovered  by  Brewster  that  com- 
plete polarization  occurs  when  the  angle  of  incidence  is  such 
that  the  reflected  ray  and  the  refracted  ray  are  at  right  angles 
to  each  other,  or  when  the  tangent  of  the  polarizing  angle  is 
equal  to  the  index  of  refraction.  The  planes  of  polarization  in 
the  reflected  and  refracted  beams  are  perpendicular  to  each 
other.  When  the  polarization  is  complete  in  the  reflected  beam, 
the  polarization  is  a  maximum  in  the  refracted  beam.  By 
making  certain  suppositions  regarding  the  relations  of  the 
components  of  the  vibrations  at  the  reflecting  surface,  which 
were  not  entirely  in  accord  with  mechanical  principles, 
although  he  treated  the  vibrations  as  if  they  were  the  vibr?- 
tions  of  some  sort  of  matter,  and  as  if  they  conformed  to  the 
law  of  conservation  of  mechanical  energy,  Fresnel  was  able  to 
show  that  light  transmitted  by  vibrations  occurring  in  the 
plane  of  incidence  will  not  be  reflected,  if  the  angle  of  incidence 
•  is  the  polarizing  angle,  but  will  be  entirely  refracted.  The 
effect  of  reflection  at  the  polarizing  angle  upon  common  light 
is  therefore  to  reflect  the  component  of  its  elliptic  vibration 
which  is  perpendicular  to  the  plane  of  incidence,  while  the 
other  component,  lying  in  the  plane  of  incidence,  is  contained 
only  in  the  refracted  beam.  At  any  other  angle  of  incidence 
than  the  polarizing  angle,  both  these  components  occur  in  both 
the  reflected  and  the  refracted  beams,  but  in  different  propor- 
tions, so  thf>t  those  beams  are  partially  polarized.  Fresnel's 
theory  led  to  certain  relations  between  the  intensities  of  the 
reflected  and  refracted  beams,  which  he  found  to  be  in  agree- 
ment with  observation. 

When  polarized  light  was  first  studied  in  connection  with 
double  refraction,  the  plane  of  polarization  was  specified  by 


LIGHT.  236 

reference  to  a  definite  plane  in  the  crystal.  A  typical  crystal 
of  Iceland  spar  is  a  rhombohedron,  bounded  by  six  equal  faces, 
each  of  which  is  a  similar  rhombus.  The  line  drawn  from  one 
obtuse  angle  of  this  rhombohedron  to  the  opposite  obtuse  angle 
is  called  the  axis  of  the  crystal.  It  is  the  line  marking  the 
direction  along  which  no  double  refraction  takes  place.  We 
have  already  called  it  the  optic  axis.  Any  plane  perpendicular 
to  one  of  the  faces  of  the  crystal  and  parallel  to  the  optic  axis 
is  called  a  principal  plane.  The  ordinary  ray  emerging  from 
the  crystal  was  conventionally  said  to  be  polarized  in  the  prin- 
cipal plane.  The  extraordinary  ray,  which  is  polarized  oppo- 
sitely to  the  ordinary  ray,  was  then  said  to  be  polarized  in  a 
plane  at  right  angles  to  the  principal  plane.  When  a  polarized 
ray  falls  upon  a  reflecting  surface  at  the  polarizing  angle  and 
in  such  a  way  that  its  plane  of  polarization  is  perpendicular 
to  the  plane  of  incidence,  it  is  not  reflected.  According  to 
Fresnel's  hypothesis,  therefore,  its  vibrations  are  in  the  plane 
of  incidence.  Fresnel  concluded,  therefore,  that  the  vibrations 
of  the  polarized  ray  are  perpendicular  to  the  plane  of  polar- 
ization. 

By  modifying  one  of  Fresnel's  hypotheses,  F.  Neumann  and 
McCulIagh  developed  a  theory  of  polarized  light  which  differed 
from  that  of  Fresnel  in  supposing  that  the  vibrations  are  in 
the  plane  of  polarization.  Both  theories  were  able  to  account 
for  all  known  facts  of  observation.  For  many  years  they  stood 
as  alternative  theories.  They  have  been  explained  and  recon- 
ciled by  the  electromagnetic  theory  of  light. 

Fresnel  was  able  to  explain  the  double  refraction  in  Iceland 
spar  and  to  develop  a  general  theory  of  double  refraction, 
which  applies  to  all  sorts  of  crystals,  on  the  hypothesis  of 
transverse  vibrations.  To  do  this  he  studied  the  effect  of  a 
disturbance  set  up  inside  a  body,  like  a  crystal,  in  which  the 
elasticity  is  different  in  different  directions.  He  showed  that 
any  general  elliptic  vibration  will  be  »t  oncn  resolved  into 
two  vibrations  at  right  angles  to  each  other,  and  that  these  will 
be  transmitted  in  different  directions  in  the  crystal  with  differ- 
ent velocities.  The  surface  reached  at  any  instant  by  the  vibra- 


236  LIGHT. 

tions  which  pass  out  in  all  directions  from  the  disturbed  centre 
is  what  is  called  the  wave  surface.  In  most  crystals,  the  wave 
surface  is  a  complicated  one,  formed  of  two  sheets,  which 
touch  each  other  at  four  symmetrically  arranged  points.  These 
points  may  be  connected  in  pairs  by  two  lines  which  pass 
through  the  centre.  These  lines  are  optic  axes,  and  a  ray  trav- 
elling along  either  one  of  them  will  not  be  doubly  refracted. 
For  certain  classes  of  crystals,  of  which  Iceland  spar  is  an 
example,  the  wave  surface  reduces  to  a  sphere  and  an  ellipsoid, 
which  touch  each  other  at  the  ends  of  one  of  the  axes  of  the 
ellipsoid.  The  line  joining  these  points  of  contact  is  the  optic 
axis.  Crystals  of  this  sort  are  called  uniaxial  crystals;  crys- 
tals of  the  other  sort  are  biaxial. 

By  the  use  of  the  Fresnel  wave  surface,  the  directions  of 
the  two  rays  formed  by  double  refraction  may  be  calculated 
for  different  angles  of  incidence,  and  the  theory  can  be  thus 
tested.  The  most  minute  observation  has  found  no  point  of 
disagreement  between  the  conclusions  of  the  theory  and  the 
results  of  observation.  The  form  of  the  wave  surface  in  crys- 
tals is  thus  proved  to  be  that  developed  by  Fresnel.  This  re- 
sult does  not,  however,  confirm  the  hypotheses  from  which 
Fresnel  deduced  it.  Other  forms  of  the  theory  lead  to  the  same, 
or  practically  to  the  same,  form  of  the  wave  surface. 

The  hypothesis  of  transverse  vibrations  received  additional 
confirmation  from  its  success  in  explaining  certain  phenomena 
discovered  by  Arago.  Arago  found  that  if  a  parallel  beam  of 
polarized  light  was  received  upon  a  crystal  so  placed  as  to 
extinguish  the  beam,  the  light  could  be  made  to  reappear  by 
interposing  in  the  path  of  the  beam  a  thin  sheet  of  mica.  The 
light  which  thus  appeared  was  colored,  and  the  color  changed 
as  the  crystal  through  which  it  was  observed  was  rotated. 
To  show  this  experiment  we  use  two  crystals  of  Iceland  spar, 
called  Nicol's  prisms.  A  Nicol's  prism  is  a  long  prism  of  Ice- 
land spar  which  has  been  cut  through  diagonally  from  one 
obtuse  angle  to  the  other,  and  cemented  together  again  with 
Canada  balsam  after  the  new  faces  have  been  polished.  A 
beam  of  light  which  falls  on  one  end  of  this  crystal  is  divided 


LIGHT.  237 

into  two  oppositely  polarized  beams,  which  proceed  through 
the  crystal  till  they  meet  the  surface  of  the  Canada  balsam. 
If  the  crystal  has  been  divided  in  the  proper  way,  the  ordi- 
nary ray  will  meet  the  balsam  at  an  angle  greater  than  its 
critical  angle.  It  will  consequently  be  totally  reflected  and 
will  not  emerge  from  the  other  end  of  the  crystal.  The  ex- 
traordinary ray,  on  the  contrary,  meets  the  balsam  within  its 
critical  angle,  and  is  therefore  partly  transmitted  through  it, 
and  emerges  from  the  crystal.  The  Nicol's  prism  may  be  thus 
used  to  obtain  a  beam  of  polarized  light.  In  trying  Arago's 
experiment,  one  Nicol's  prism  is  used  as  the  polarizer.  The 
other,  called  the  analyser,  is  placed  in  the  path  of  the  polar- 
ized beam  and  the  light  which  comes  through  it  is  observed 
either  on  a  screen  or  by  the  eye.  When  the  analyser  is 
turned  so  that  its  principal  plane  is  at  right  angles  to  that 
of  the  polarizer,  no  light  passes  through  it.  If  we  now  inter- 
pose between  the  two  prisms  a  sheet  of  mica,  colored  light 
passes  through  the  analyser.  In  only  two  positions  of  the 
mica  sheet  will  this  not  occur,  when  its  principal  plane  is 
either  parallel  with,  or  perpendicular  to,  the  plane  of  polar- 
ization of  the  incident  beam.  For  other  positions  of  the  mica, 
no  position  of  the  analyser  can  be  found  in  which  light  will 
not  pass  through  it.  Fresnel  explained  this  phenomenon  in  the 
following  way:  The  vibrations  of  the  polarized  beam  which 
loll  on  the  mica  are  resolved  into  two  components  perpendicu- 
lar to  each  other.  The  velocities  of  these  components  are  dif- 
ferent, and  when  they  emerge  from  the  mica  one  of  them  has 
gained  a  fraction  of  a  wave  length  on  the  other,  so  that  they  are 
then  in  different  phases.  Because  the  wave  lengths  of  different 
colors  are  different,  the  differences  in  ph"ase  between  the  two 
emergent  beams  are  different  for  the  different  colors.  After 
emergence,  these  beams  proceed  to  the  analyser,  which  resolves 
each  of  them  into  two  perpendicular  components,  and  transmits 
one  component  of  each.  The  components  transmitted  are  par- 
allel with  each  other,  and  may  therefore  interfere.  If  they 
happen  to  be  in  opposite  phases,  they  destroy  each  other.  If 
they  are  in  the  same  phase  they  enhance  each  other.  For  a 


238  LIGHT. 

given  thickness  of  the  mica  some  of  the  constituents  of  the 
original  white  beam  will  thus  be  destroyed,  and  the  light  which 
will  pass  will  appear  colored.  The  light  which  is  rejected  by 
the  analyser  contains  those  colors  which  are  absent  in  the 
transmitted  light,  so  that  if  the  analyser  is  turned  so  that  it 
transmits  the  light  which  it  formerly  rejected,  the  comple- 
mentary color  appears. 

Similar  effects  may  be  produced  by  the  use  of  thin  sheets  of 
crystals  other  than  mica. 

By  using  divergent  or  convergent  light  instead  of  parallel 
light,  very  complicated  systems  of  colored  figures  may  be  pro- 
duced. The  peculiarities  of  these  systems  may  be  calculated 
from  a  knowledge  of  the  optical  properties  of  the  particular 
crystal  which  is  interposed  in  the  beam.  When  the  results  of 
calculation  are  compared  with  the  results  of  observation,  a  very 
complete  agreement  is  found  between  them.  This  general  result 
furnishes  additional  confirmation  of  the  Fresnel  wave  surface. 

Arago  discovered  a  peculiar  effect  produced  by  quartz  on 
polarized  light,  which  was  also  explained  by  Fresnel.  Quartz 
is  a  uniaxial  crystal,  and  light  falling  perpendicularly  upon  a 
plate  of  quartz  cut  perpendicularly  to  the  optic  axis  is  not 
doubly  refracted.  If,  however,  an  analyser  is  so  placed  in  a 
polarized  beam  as  to  extinguish  it,  and  if  then  such  a  plate  of 
quartz  is  interposed  in  the  polarized  beam,  light  will  again 
come  through  the  analyser.  Unless  the  plate  of  quartz  is  too 
thick,  this  light  is  colored.  It  is  not  altered  by  rotation  of  the 
quartz  around  the  beam  as  an  axis,  but  when  the  analyser  is 
rotated,  the  color  changes  continually.  If  light  of  one  color  is 
used  in  this  experiment,  it  will  pass  through  the  analyser  when 
the  quartz  plate  is  introduced,  and  if  the  analyser  is  then 
turned  through  a  certain  angle,  which  is  different  for  each 
color,  the  light  will  be  extinguished.  From  this  experiment  it 
is  easy  to  see  that  the  different  colors  which  appear,  when 
white  light  is  used,  are  due  to  the  suppression  of  some  of  the 
constituents  of  white  light  in  each  position  of  the  analyser. 
This  action  of  quartz  is  known  as  the  rotation  of  the  plane  of 
polarization.  Many  other  substances,  among  them  solutions  of 


sugar  and  of  other  organic  bodies,  were  subsequently  found  to 
possess  the  same  property.  From  the  fact  that  it  is  possessed 
by  solutions,  in  \vhich  it  is  impossible  to  suppose  that  the  act- 
ive molecules  have  any  definite  directions,  such  as  they  may  be 
supposed  to  have  in  crystals,  we  conclude  that  this  action  on 
polarized  light  is  due  to  the  structure  of  the  molecule  itself, 
or  perhaps  of  one  of  its  atoms.  Fresnel  explained  this  phe- 
nomenon by  supposing  that  the  polarized  ray,  on  entering  the 
quartz,  sets  up  two  circular  vibrations  in  opposite  senses.  It 
is  easy  to  see  that  the  resultant  of  two  such  vibrations  will  be 
a  rectilinear  vibration  like  that  of  the  incident  ray.  He  further 
supposed  that  the  velocity  in  the  crystal  of  one  of  these  vibra- 
tions is  greater  than  that  of  the  other.  If  this  is  the  case,  the 
two  circular  vibrations,  on  their  emergence  from  the  quartz, 
will  combine  again  to  produce  a  rectilinear  vibration,  which  is 
inclined  to  the  one  which  entered  the  quartz  by  an  amount 
proportional  to  the  thickness  of  the  plate.  If  the  circular 
vibration  which  is  traveling  faster  is  the  one  in  which  the 
rotation  is  clock-wise  to  an  observer  looking  along  the  ray,  the 
plane  of  polarization  of  the  emergent  light  will  be  turned  clock- 
wise. If  the  other  circular  vihrnti"ii  travels  faster  through  the 
quartz,  the  plane  of  polarization  will  be  turned  counter-clock- 
wise. Specimens  of  quartz  are  found  which  show  each  of  these 
rotations.  They  are  called  right-handed  and  left-handed, 
respectively.  By  an  ingenious  combination  of  quartz  prisms 
properly  cut,  Fresnel  was  able  to  separate  the  two  circularly 
polarized  rays  assumed  by  him  in  this  explanation,  and  so  to 
prove  its  correctness. 

In  his  study  of  the  reflection  of  polarized  light  Fresnel  was 
led  to  consider  the  effect  of  total  reflection  upon  the  plane  of 
polarization.  In  that  case  he  found  that  when  the  vibration  in 
the  polarized  beam  is  not  at  right  angles  to  the  plane  of  inci- 
dence, that  component  of  it  which  lies  in  the  plane  of  incidence 
has  its  phase  reversed  by  the  reflection.  In  general  the  com- 
bination of  the  new  component  thus  produced  with  the  other 
component  which  is  reflected  without  change  produces  an  ellip- 
tic vibration.  By  combining  two  such  total  reflections,  taking 


240  LIGHT. 

place  at  the  proper  angles,  the  vibration  of  the  reflected  beam 
becomes  circular.  The  beam  is  then  said  to  be  circularly  polar- 
ized. Circular  polarization  may  also  be  produced  by  the  use 
Of  a  sheet  of  mica,  whose  thickness  is  such  that  one  of  the  two 
rays  formed  in  it  by  double  refraction  gains  a  quarter  of  a 
wave  length  on  the  other  in  its  passage  through  the  sheet.  This 
sheet  is  placed  in  the  path  of  a  polarized  beam  in  such  a  posi- 
tion that  its  principal  plane  makes  an  angle  of  45°  with  the 
plane  of  polarization.  In  this  case  the  two  components  which 
emerge  from  it  contain  vibrations  of  equal  magnitude  and  dif- 
fering in  phase  by  one  quarter  of  a  period.  These  two  vibra- 
tions combine  to  produce  a  circular  vibration.  Such  a  sheet 
of  mica  is  called  a  quarter-wave  plate. 

When  polarized  light  is  incident  obliquely  upon  a  polished 
metallic  surface,  the  reflected  light  is  elliptically  polarized. 

It  was  discovered  by  Faraday  that  when  a  beam  of  plane 
polarized  light  traverses  a  magnetic  field  in  the  direction  of 
its  lines  of  force,  the  plane  of  polarization  is  rotated.  The 
amount  of  rotation  depends  upon  the  strength  of  the  field  and 
upon  the  substance  through  which  the  light  is  passing.  It  was 
found  by  Faraday  first  in  a  glass  of  a  peculiar  composition. 
The  rotation  produced  by  the  magnetic  field  differs  from  that 
produced  by  quartz  in  one  important  respect.  If  the  beam 
which  has  been  rotated  by  the  magnetic  field  is  received  on  a 
plane  mirror  and  sent  back  through  the  field  again,  it  under- 
goes an  additional  rotation  in  the  same  sense.  On  the  other 
hand,  a  beam  which  is  sent  back  through  the  quartz  undergoes 
rotation  in  the  opposite  sense,  so  that  its  plane  of  polarization 
becomes  the  same  as  that  of  the  incident  beam.  Magnetic  rota- 
tion of  the  plane  of  polarization  is  explained  in  the  same  gen- 
eral way  as  the  rotation  by  quartz.  The  two  circularly  polar- 
ized beams  have  recently  been  separated  by  Brace. 

149.  Spectrum  Analysis. — When  salts  of  the  elements  are 
introduced  into  a  flame,  they  give  the  flame  characteristic 
colors.  For  example,  sodium  chloride  or  sodium  carbonate  will 
color  the  flame  yellow.  If  the  light  from  this  flame  passes 
through  a  narrow  slit  and  is  received  on  a  prism  so  as  to  pro- 


LIGHT.  241 

duce  a  pure  spectrum,  certain  parts  of  the  spectrum  are  found 
to  be  much  more  intense  than  the  rest  of  it.  In  the  case  of  the 
sodium  salts  a  narrow  band  or  line,  very  brilliantly  illumin- 
ated, appears  in  the  yellow.  This  yellow  line  of  sodium  was 
noticed  by  Herschel.  By  the  investigations  of  Bunsen  and 
Kirchhoff,  1859-1862,  it  was  shown  to  be  characteristic  of  the 
presence  of  sodium  vapor  in  the  flame,  so  that  whenever  this 
line  can  be  detected,  it  may  be  inferred  that  sodium  is  present. 
Bunsen  studied  the  characteristic  lines,  or,  as  we  may  say,  the 
spectra  of  different  elements,  and  showed  that  they  can  be  used 
as  a  means  of  detecting  the  presence  of  those  elements  in  the 
substance  by  which  the  flame  is  colored.  Elements  which  can 
not  be  vaporized  in  a  flame  may  be  vaporized  by  the  heat  of  the 
electric  arc,  and  will  then  give  similar  characteristic  line  spec- 
tra. When  the  electric  spark  is  passed  through  an  elementary 
gas,  a  similar  characteristic  spectrum  is  produced. 

This  method  of  detecting  the  presence  of  a  particular  ele- 
ment in  a  compound  by  means  of  its  characteristic  spectrum, 
is  called  spectrum  analysis. 

If  the  spectrum  of  a  gas  is  observed  as  the  pressure  upon 
it  is  increased,  it  is  found  that  the  lines,  which  at  first  are 
sharp  and  narrow,  gradually  broaden  out  into  bands,  and  as 
the  pressure  is  still  further  increased,  these  bands  overlap, 
until  the  spectrum  becomes  continuous,  like  that  of  an  incan- 
descent solid. 

150.  Fraunhofer's  Lines. — When  a  pure  spectrum  is  formed 
with  sunlight,  it  is  found  to  be  crossed  by  a  great  number  of 
dark  lines.  Some  of  the  most  intense  of  these  lines  were  ob- 
served by  Wollaston,  and  they  were  more  accurately  studied  by 
Fraunhofer.  The  light  from  an  ordinary  flame  or  from  an 
incandescent  body  does  not  show  these  lines. 

It  was  noticed  by  several  observers  that  one  of  the  most 
conspicuous  of  the  Fraunhofer  lines,  in  the  yellow  light  of  the 
spectrum,  coincided  in  position  with  the  yellow  line  of  the 
spectrum  of  sodium.  This  coincidence  was  verified  by  the  exact 
observations  of  Kirchhoff,  who  was  working  with  Bunsen  on 
the  development  of  spectrum  analysis.  Kirchhoff  suspected 


242  LIGHT. 

that  the  dark  line  in  the  solar  spectrum  is  due  to  the  absorp- 
tion of  light,  coming  from  the  central  part  of  the  sun,  by  the 
vapor  of  sodium  in  its  outer  atmosphere.  To  show  that  this 
may  be  the  case,  he  observed  the  spectrum  of  the  white  light 
coming  from  incandescent  lime,  heated  by  the  oxyhydrogen 
flame,  when  a  flame  containing  sodium  vapor  was  placed  in 
front  of  the  slit.  He  found,  with  those  conditions,  that  a  dark 
line  appeared  in  the  spectrum,  coinciding  in  position  with  the 
bright  yellow  line  which  the  sodium  vapor  itself  would  have 
given.  He  explained  the  production  of  this  dark  line  by  the  aid 
of  the  principle  of  resonance.  From  the  fact  that  sodium  vapor 
emits  light  of  a  certain  period,  as  shown  by  its  giving  rise  to 
the  characteristic  yellow  line  of  its  spectrum,  it  is  evident  that 
those  elements  of  sodium  vapor  which  emit  light  execute  vibra- 
tions of  that  period.  Now,  if  light  of  all  periods  of  vibration 
falls  upon  the  sodium  vapor,  those  vibrations  which  are  not 
similar  to  the  natural  vibrations  of  the  sodium  will  pass  on 
without  disturbing  it,  but  those  vibrations  which  have  the  same 
period  as  that  of  the  sodium  will  increase  its  natural  vibration 
by  giving  to  it  impulses  properly  timed,  and  so  will  them- 
selves be  diminished  in  intensity,  or  absorbed.  The  dark  line 
which  is  thus  cast  in.the  spectrum  is  not  black,  but  simply  not 
so  intensely  illuminated  as  the  regions  on  either  side  of  it. 

A  comparison  of  the  Fraunhofer  lines  of  the  solar  spectrum 
with  the  spectra  of  the  various  elements  led  to  the  discovery  of 
many  coincidences  similar  to  that  described  in  the  case  of 
sodium.  In  some  cases,  as  in  that  of  iron,  for  example,  these 
coincidences  extend  to  hundreds  of  lines.  In  every  case  in 
which  such  coincidences  can  be  proved,  we  infer  the  presence 
of  the  particular  element  giving  the  spectrum  in  the  outer 
fitmosphere  of  the  sun. 

When  a  colored  transparent  body  is  placed  in  the  path  of  a 
beam  which  forms  a  spectrum,  there  often  appear  particular 
portions  of  the  spectrum  in  which  the  light  is  less  intense  than 
it  was  before.  These  darker  regions  are  called  absorption 
bands.  They  are  characteristic  of  the  particular  substance 
which  intercepts  the  light,  and  may  be  used  as  a  means  of 
analysis. 


LIGHT.  243 

Kirchhoff  found  by  a  general  theoretical  investigation  that 
the  absorption  of  yellow  light  by  sodium  vapor  is  an  example 
of  a  perfectly  general  law,  that  any  body  will  absorb  those 
waves  of  light  which  it  will  itself  emit  when  self-luminous. 
The  ratio  between  the  emissive  power  and  the  absorptive  power 
is  the  same  for  all  substances  at  the  same  temperature.  Those 
bodies  which  absorb  all  colors,  also  emit  all  colors  when  they 
become  self-luminous.  Those  bodies  which  absorb  only  partic- 
ular colors,  emit  only  those  particular  colors. 

We  are  now  in  a  position  to  consider  the  question  of  the 
color  of  bodies.  All  bodies  which  are  not  self-luminous  are 
seen  by  means  of  the  light  reflected  by  them,  coming  from  the 
sun,  or  from  some  other  self-luminous  source.  Very  many 
bodies  show  the  same  color  when  examined  by  transmitted 
light  as  when  examined  by  reflected  light.  To  explain  their 
color,  we  suppose  that  some  of  the  constituents  of  the  white 
light  which  falls  upon  them  are  absorbed,  and  that  the  light 
which  is  reflected  to  the  eye,  and  by  which  the  body  is  seen,  has 
penetrated  sufficiently  within  the  body  to  allow  this  absorption 
to  deprive  it  of  those  constituents.  We  therefore  see  it  really 
by  light  which  has  traversed  enough  of  thce  body  for  absorption 
to  have  its  full  effect. 

There  are  many  other  bodies,  however,  for  which  the  color 
in  the  transmitted  light  is  different  from  that  in  the  reflected 
light.  With  them  the  reflection  seems  to  occur  at  the  surface. 
It  is  found  in  every  such  case  that  absorption  bands  appear  in 
the  transmitted  light,  and  that  the  light  which  is  reflected  is 
exactly  that  which  is  wanting  in  the  transmitted  beam.  These 
bodies  show  also  another  peculiarity,  which  was  discovered  by 
Christiansen  in  fuchsine,  and  which  was  fully  studied  by 
Kundt.  This  peculiarity  i?  oallod  anomalous  dispersion.  In 
very  many  cases,  the  spectrum  formed  by  a  prism  of  a  particu- 
lar substance,  such  as  glass,  for  example,  has  the  colors 
arranged  in  the  order  of  their  wave  lengths.  A  dispersion  of 
this  sort  is  supposed  to  be  according  to  law,  and  any  deviation 
from  it  is  anomalous.  When  light  is  sent  through  a  prism  of 
the  substance  showing  anomalous  dispersion,  the  order  of  the 


244  LIGHT. 

colors  of  the  spectrum  is  not  the  order  of  the  wave  lengths.  It 
was  found  by  Kundt  to  be  a  general  law  that  the  colors  which 
are  displaced  from  their  position  in  the  ordinary  spectrum  are 
those  which  lie  on  either  side  of  an  absorption  band.  The  color 
whose  wave  length  is  longer  is  displaced  toward  the  violet  end 
of  the  spectrum,  and  that  whose  wave  length  is  shorter,  toward 
the  red  end. 

All  these  peculiarities  of  substances  which  show  surface 
color  and  anomalous  dispersion  can  be  explained  by  supposing 
that  the  elements  of  their  structure  which  can  emit  light  have 
vibrations  of  their  own  of  the  same  period  as  the  light  which 
the  substance  absorbs. 

151.  The  Extent  of  the  Spectrum. — Herschel  examined  the 
heating  effect  produced  by  the  different  parts  of  the  spectrum 
by  placing  a  thermometer  in  it;  and  found  that  as  the  bulb  of 
the  thermometer  was  moved  down  toward  the  red  end  of  the 
spectrum  the  heating  effect  became  more  and  more  pronounced, 
and  that  a  still  greater  heating  effect  appeared  when  the  bulb 
was  placed  in  the  region  just  beyond  the  red  end  of  the  spec- 
trum. He  concluded  that  rays  exist  of  longer  wave  length  than 
the  red  rays.  When  the  action  of  light  on  salts  of  silver  was 
discovered,  on  which  photographic  processes  depend,  it  was 
found  that  the  most  active  region  in  this  respect  lay  outside  the 
violet  end  of  the  spectrum.  The  invisible  rays  which  produced 
the  heating  effect  were  at  first  called  heat  rays,  and  those  which 
produced  the  chemical  effect,  actinic  rays;  but  it  is  easily  seen 
that  there  is  no  reason  for  considering  them  to  be  essentially 
different  from  the  visible  rays.  Accordingly  it  has  been  found 
that  all  these  rays,  withoxit  exception,  produce  a  heating  effect, 
and  that  the  chemical  effect  has  been  caused  by  so  many  of 
them,  that  the  conclusion  is  fully  warranted  that  the  rays 
from  a  luminous  body  or  from  any  body  are  of  the  same  nature, 
and  differ  only  in-  the  lengths  of  the  waves  whose  direction  of 
transmission  they  indicate.  The  examination  of  the  radiation 
from  incandescent  vapors  shows  that  many  spectral  lines  are 
emitted  by  them  which  lie  in  the  invisible  parts  of  the  spec- 
trum. And  similarly,  multitudes  of  Fraunhofer  lines  are  de- 
tected in  the  invisible  parts  of  the  solar  spectrum. 


LIGHT.  246 

The  shortest  waves  in  the  extreme  violet  are  about  0.0004 
millimetres  long;  the  longest  in  the  extreme  red,  about  0.0007 
millimetres  long.  Photographic  methods  have  detected  waves 
0.0001  millimetres  long,  and  observations  of  the  heating  effect 
have  detected  waves  0.02  millimetres  long. 

Balmer  showed  that  the  spectral  lines  of  hydrogen  are  so 
disposed  that  their  wave  lengths,  or  the  number  of  vibrations 
corresponding,  can  be  calculated  by  the  aid  of  a  general  form- 
ula. In  applying  this  formula  each  spectrum  line  is  assigned 
one  of  the  natural  numbers,  according  to  its  position  in  the 
hydrogen  spectrum,  beginning  with  3,  assigned  to  the  line  corre- 
sponding to  the  longest  wave  length,  and  the  insertion  in  the 
formula  of  the  number  assigned  ip  the  line  leads  to  the  number 
of  vibrations  corresponding  to  the  line.  Similar  series,  calcu- 
lable by  similar,  though  more  complicated  formulae,  have  been 
discovered  in  the  spectra  of  many  other  elements. 

152.  Fluorescence. — Certain  substances,  such  as  fluor-spar 
or  solutions  of  chlorophyll  or  of  sulphate  of  quinine,  when 
placed  so  as  to  receive  a  narrow  beam  of  light,  become  self- 
luminous  in  a  peculiar  way.    The  light  emitted  by  them  seems 
to  originate  in  the  path  of  the  beam,  and  has  a  characteristic 
color  and  spectrum,  depending  on  the  nature  of  the  substance. 
The  phenomenon  is  known  as  fluorescence.    According  to  Stokes 
the  wave  length  of  the  emitted  light  is  always  less  than  that  of 
the  light  which  excites  it.     This  law  has  recently  been  dis- 
proved by  Nichols  and  Merritt.     The  production  of  the  light 
may  be  explained  as  a  species  of  resonance. 

By  rapidly  cutting  off  and  renewing  the  incident  beam,  it 
has  been  shown  that  the  emitted  light  persists,  though  often  for 
only  a  short  time,  after  the  incident  beam  is  cut  off.  Fluor- 
escence is  therefore  apparently  not  distinct  from  phospho- 
rescence, that  is,  from  the  phenomenon  of  persistent  lumin- 
escence excited  by  exposure  to  light  in  certain  substances,  such 
as  sulphide  of  calcium. 

153.  Zeeman  Effect. — It  has  recently  been  discovered  by 
Zeeman  that  the  vibrations  of  a  vapor  which  emits  light  are 
peculiarly  modified  if  they  are  executed  in  a  magnetic  field. 


24G  LIGHT. 

We  need  not  describe  this  modification  further  than  to  say, 
that  a  single  spectral  line  which  the  vapor  would  ordinarily 
emit  is  divided,  when  the  vapor  is  in  a  magnetic  field,  into  two 
or  more  lines,  and  that  the  light  in  these  lines  is  differently 
polarized. 

We  are  not  able  to  explain  the  Zeeman  effect,  nor  indeed  to 
give  an  adequate  explanation  of  the  hypotheses  upon  which  our 
whole  theory  of  light  has  been  based,  by  any  purely  mechani- 
cal theory  of  the  ether  and  of  the  nature  of  light.  It  is  now 
quite  certain  that  what  we  have  called  the  vibrations  of  light 
are  periodic  electric  disturbances  in  the  ether,  and  that  the 
various  modifications  impressed  upon  them  by  material  bodies 
are  due  to  the  electric  relations  of  those  bodies.  WTe  shall  dis- 
cuss the  electromagnetic  theory  of  light  in  connection  with  our 
study  of  electricity. 

154.  Velocity  of  Light. — The  first  direct  determination  of 
the  velocity  of  light,  by  a  method  which  did  not  involve  astro- 
nomical measurements,  was  made  by  Fizeau  in  1849.  He  used 
a  toothed  wheel  which  could  be  rapidly  revolved  at  a  known 
rate.  When  the  wheel  was  still,  a  beam  of  light  was  sent  out 
through  one  of  the  gaps  between  the  teeth,  was  received  on  a 
mirror  set  up  at  a  great  distance,  and  reflected  back  through 
the  same  gap.  The  wheel  was  then  set  in  rotation.  At  first, 
as  it  turned  slowly,  the  light  sent  back  from  the  mirror  still 
passed  through  the  same  gap,  but  as  the  rotation  was  increased, 
the  neighboring  tooth  moved  forward  into  the  path  of  the 
light,  so  as  to  intercept  it.  When  the  rotation  of  the  wheel  was 
doubled,  the  returning  light  passed  through  the  next  gap.  In 
this  way,  by  increasing  the  rate  of  rotation,  the  returning  light 
was  alternately  intercepted  and  transmitted.  By  observing  the 
different  rates  of  rotation,  it  was  easy  to  determine  the  time 
occupied  by  the  light  in  passing  from  the  wheel  to  the  mirror 
and  back  again  to  the  wheel.  The  velocity  of  light  determined 
in  this  way  was  found  to  be  about  315  million  metres  per 
second. 

Another  method  of  determining  the  velocity  of  light  was 
employed  by  Foucault,  in  1862.  He  used  for  that  purpose  a 


LIGHT.  247 

small  mirror,  which  could  be  rotated  rapidly  around  an  axis 
parallel  to  its  face.  When  the  mirror  was  at  rest,  light  was 
allowed  to  fall  on  it  from  a  slit,  and  was  reflected  by  it  to  a 
fixed  mirror.  This  mirror  sent  the  light  back  over  the  same 
path,  so  that  it  fell  on  the  slit.  When  the  mirror  was  rotated 
the  angle  at  which  it  received  the  returning  light  was  different 
from  that  at  which  it  had  sent  out  the  same  light  from  the 
slit,  and  the  returning  light  reflected  from  it  no  longer  fell  on 
the  slit.  The  velocity  of  light  was  determined  by  measuring  the 
displacement  of  the  image  of  the  slit  sent  back  from  the  revolv- 
ing mirror  and  the  rate  of  rotation  of  the  mirror.  Foucault 
obtained  a  result  for  the  velocity  of  light  which  was  a  little 
smaller  than  that  obtained  by  Fizeau.  This  method  has  been 
used  by  Michelson  and  by  Newcomb  to  obtain  the  velocity  of 
light  which  is  now  considered  the  standard.  Newcomb  gives  it 
as  299,860  kilometres  per  second.  For  ordinary  purposes  we 
may  take  it  as  equal  to  300  million  metres  per  second. 

It  was  by  the  use  of  this  method  that  Foucault  proved,  in 
1850,  that  the  velocity  of  light  in  water  is  less  than  in  air. 

155.  Effect  on  the  Velocity  of  Light  of  the  Motion  of 
Bodies. — When  the  wave  theory  of  light  was  first  studied  by 
Fresnel  and  Arago,  the  question  of  the  aberration  of  light  had 
to  be  considered."  Arago  perceived  that,  on  the  accepted  expla- 
nation of  aberration,  the  aberration  of  a  star  ought  to  be  dif- 
ferent from  that  ordinarily  obtained  for  it3  if  the  tube  of  the 
telescope  by  which  it  was  observed  was  filled  with  water.  Ob- 
servation showed,  however,  that  the  aberration  was  the  same 
when  thus  determined  as  it  had  previously  been  found  to  be.  It 
was  shown  by  Fresnel  that  this  result  could  be  explained  if  the 
velocity  of  light  was  supposed  to  be  affected  by  the  velocity  of 
the  body  through  which  it  was  passing,  according  to  a  certain 
law.  In  order  to  test  this  hypothesis,  Fizeau  observed  the 
change  produced  in  the  velocity  of  light  by  sending  it  through 
a  stream  of  water.  The  results  of  his  observations,  which  have 
since  been  confirmed  by  Michelson  and  Morley,  were  in  agree- 
ment with  Fresnel's  formula. 


248  LIGHT. 

These  experiments  are  consistent  with  the  hypothesis  that  the 
ether  is  at  rest,  and  does  not  share  in  the  motion  of  bodies 
moving  through  it.  On  the  other  hand,  Michelson  and  Morley,  by 
the  use  of  an  instrument  called  the  interferometer,  compared  the 
velocity  of  light  in  the  direction  of  the  earth's  motion  with  that  of 
light  perpendicular  to  the  earth's  motion,  and  found  that  they 
could  not  detect  any  difference,  such  as  would  be  expected  if  the 
ether  is  at  rest.  This  result  is  at  variance  with  most  of  the  other 
facts  known  bearing  on  the  subject,  and  while  it  is  accepted  as 
proved,  the  conclusion  that  the  ether  moves  with  the  earth  is  not 
generally  drawn. 

The  question  of  the  action  of  moving  matter  upon  the  ether, 
and  of  the  way  in  which  the  motion  of  matter  affects  the  prog- 
ress of  light  through  it  has  not  yet  been  solved.  Such  explana- 
tions as  have  been  given  depend  upon  certain  assumed  electric 
properties  of  the  ether  and  of  matter,  and  will  be  considered  in 
connection  with  our  study  of  electricity. 


MAGNETISM.  249 


MAGNETISM. 

156.  The  Lodestone. — From  the  earliest  antiquity  it  has 
been  known  that  certain  stones,  or  pieces  of  mineral,  exist 
which  possess  the  peculiar  property  of  attracting  iron.  These 
stones  are  called  magnets  or  lodestones.  In  view  of  the  fact 
that  pieces  of  iron  or  steel  artificially  prepared,  and  possessing 
this  same  property,  are  also  called  magnets,  it  may  be  best 
always  to  designate  these  natural  magnets  as  lodestones.  They 
contain  principally  magnetic  oxide  of  iron. 

The  attractive  power  of  the  lodestone  for  iron  is  shown 
more  strongly  at  some  parts  of  it  than  at  others.  By  careful 
selection,  lodestones  may  be  found,  in  which  the  regions  of 
strongest  attraction  are  two  in  number,  but  none  are  ever  found 
with  only  one  such  region. 

It  was  found  by  the  early  observers  that  when  a  small  piece 
of  iron,  such  as  an  iron  finger  ring,  was  attracted  by  the  lode- 
stone,  it  also  acquired  the  property  of  attracting  iron.  The 
iron  thus  attracted  by  it  acquired  in  its  turn  the  same  property 
of  attraction.  The  attractive  force  developed  in  each  successive 
piece  of  iron  decreased  in  intensity  as  the  iron  was  further 
removed  from  the  original  lodestone.  These  pieces  of  iron  are 
said  to  be  magnetized  by  induction.  Later  observations  showed 
that  iron  can  be  magnetized  by  induction  when  it  is  brought 
near  the  lodestone,  or  near  another  magnet,  without  being  in 
contact  with  it. 

When  a  lodestone  exhibiting  two  centres  of  attraction  was 
placed  in  a  light  vessel,  and  floated  on  the  surface  of  water,  it 
always  turned  about  so  that  one  of  the  two  centres  pointed 
toward  the  north,  and  the  other  toward  the  south.  This  prop- 
erty of  assuming  a  definite  direction  is  generally  better  exhib- 
ited by  artificial  magnets,  in  which  the  two  centres  of  attrac- 
tion are  more  precisely  developed,  than  it  is  by  the  lodestone. 
The  compass  is  a  magnet  so  arranged  that  it  can  turn  freely  in 
a  horizontal  plane  and  indicate  by  the  direction  in  which  it 
points  the  north  and  south  line. 


250  MAGNETISM. 

The  first  scientific  study  of  magnets  was  made  by  Gilbert, 
who  published  the  results  of  his  study  in  1600.  He  found  that 
when  the  ends  of  two  lodestones  of  the  simplest  type  were 
brought  near  each  other,  they  were  attracted  to  each  other  if 
one  of  them  was  an  end  which  would  point  toward  the  north, 
and  the  other  an  end  which  would  point  toward  the  south.  If 
ends  which  would  point  toward  the  north,  or  ends  which  would 
point  toward  the  south,  were  brought  near  each  other,  they 
repelled  each  other.  Gilbert  called  these  ends,  at  which  the 
forces  of  attraction  and  repulsion  were  most  strongly  exhib- 
ited, poles,  the  one  which  pointed  toward  the  north  being  called 
the  north  pole,  the  other,  the  south  pole.  We  may  express  Gil- 
bert's discovery  by  saying  that  a  north  pole  of  a  magnet  will 
attract  a  south  pole  of  another  magnet,  and  that  the  north 
poles  or  the  south  poles  of  two  magnets  will  repel  each  other. 

Gilbert  conceived  of  the  magnetic  condition  of  a  lodestone, 
or  of  any  magnet,  as  due  to  some  sort  of  arrangement  through- 
out its  whole  body.  The  experiment  by  which  he  was  led  to 
this  conclusion  consisted  in  cutting  a  lodestone  exhibiting  two 
poles  into  two  parts,  by  a  section  across  the  line  joining  the 
poles.  When  the  two  parts  thus  made  were  separated  and 
examined,  it  was  found  that  the  poles  which  they  originally 
possessed  had  not  been  altered,  but  that  two  new  poles  had  been 
developed,  one  in  each  piece,  of  a  sort  different  from  that 
already  in  it.  Each  piece  was  therefore  a  complete  magnet, 
having  north  and  south  poles.  When  the  newly  made  north 
and  south  poles  were  brought  near  each  other,  they  attracted 
each  other,  and  when  the  two  pieces  were  allowed  to  meet,  the 
new  poles  disappeared,  and  only  the  original  poles  remained. 
It  has  been  shown  that  whenever  a  piece  of  any  size  is  cut  off 
from  a  magnet,  it  is  always  itself  a  complete  magnet,  and  it  is 
inferred  that  if  the  molecules  of  a  magnet  could  be  separated 
from  each  other,  each  of  them  would  be  a  complete  magnet. 

157.  Artificial  Magnets. — As  has  already  been  described,  a 
piece  of  iron  may  be  made  a  magnet  by  bringing  it  near  a  lode- 
stone.  If  it  is  removed  from  the  lodestone  and  tested,  it  will  be 
found  to  be  a  magnet  still.  The  slightest  disturbance  of  it,  by 


MAONKTI>M.  251 

striking  or  jarring  it,  will  cause  it  to  lose  its  magnetic  condi- 
tion. If  the  iron  is  in  the  form  of  steel,  however,  it  will  retain 
the  magnetic  condition  induced  in  it  by  the  lodestone  to  a  very 
great  degree.  A  piece  of  steel  thus  prepared  is  called  an  arti- 
ficial magnet.  Since  its  magnetic  condition  is  retained  by  it 
indefinitely,  in  ordinary  circumstances,  it  is  also  called  a  per- 
manent magnet.  It  is  with  such  steel  magnets  that  we  can 
most  easily  carry  out  the  experiments  which  Gilbert  described 
as  carried  out  by  him  with  the  lodestone. 

The  circumstances  involved  in  magnetization  by  induction 
should  be  more  fully  examined  at  this  point.  To  do  so  we 
present  one  end  of  a  soft  iron  rod  to  the  north  pole  of  a  perma- 
nent magnet,  and  examine  the  magnetic  condition  of  the  rod. 
It  is  found  that  the  rod  is  magnetized  in  such  a  way  that  the 
end  near  the  north  pole  of  the  magnet  is  a  south  pole,  and  the 
other  end  is  a  north  pole.  If  one  end  of  the  rod  is  presented  to 
the  south  pole  of  the  magnet,  that  end  becomes  a  north  pole, 
and  the  more  distant  end  a  south  pole.  A  similar  result  is 
obtained  if  one  end  of  a  row  of  short  iron  rods  is  presented  to 
the  pole  of  the  magnet.  Each  of  the  rods  then  becomes  a  mag- 
net, with  a  pole  different  from  that  of  the  inducing  magnet  in 
the  end  nearer  it,  and  a  pole  like  that  of  the  inducing  magnet 
in  the  more  distant  end.  We  shall  subsequently  describe  the 
region  around  a  magnet,  in  which  magnetic  force  can  be  per- 
ceived by  the  aid  of  another  magnet,  as  a  magnetic  field,  and 
we  shall  then  describe  magnetic  induction  in  a  somewhat  more 
general  way.  For  the  present  it  is  sufficient  to  say,  that  a 
magnetic  pole  will  induce  a  pole  unlike  itself  in  that  part  of  a 
piece  of  iron  which  is  nearest  to  it.  The  attraction  between 
these  two  unlike  poles  draws  the  iron  and  the  magnet  together. 
It  was  this  attraction  which  was  first  observed,  and  which  was 
for  a  long  time  supposed  to  be  the  fundamental  property  of  a 
magnet.  It  was  not  until  Gilbert  examined  the  action  of  one 
magnet  on  another,  and  observed  the  magnetization  by  induc- 
tion, that  the  true  relations  of  a  magnet  to  iron  could  be  under- 
stood. 


252  MAGNETISM. 

A  piece  of  steel  may  be  made  a  magnet  by  bringing  it  in 
contact  with  a  lodestone  or  a  pole  of  another  magnet.  Various 
operations,  by  which  this  contact  is  made  according  to  certain 
rules,  have  been  devised,  in  order  to  effect  this  magnetization  as 
uniformly  and  as  powerfully  as  possible.  These  methods  have 
all  been  superseded  by  a  method  which  depends  upon  the  fact 
that  an  electric  current  sets  up  a  magnetic  field  around  itself. 
The  wire  carrying  the  current  is  wound  into  a  spiral  coil,  and 
the  steel  bar,  which  is  to  be  magnetized,  is  placed  in  the  axis 
of  the  coil.  The  magnets  produced  in  this  way  are  in  every 
way  better  than  those  produced  by  the  older  methods  of  con- 
tact. 

If  two  similar  bar  magnets  are  placed  side  by  side  with 
their  like  poles  contiguous,  they  produce  a  magnet  which  is 
more  powerful  than  either  of  them.  By  larger  combinations  of 
magnets,  very  powerful  permanent  magnets  can  be  produced. 

158.  The  Earth  as  a  Magnet. — Attention  has  already  been 
called  to  the  fact  that  a  magnet  which  is  free  to  turn  in  a  hori- 
zontal plane  will  point  with  one  of  its  ends  toward  the  north. 
The  vertical  plane  containing  the  direction  in  which  it  points 
at  any  place  is  called  the  magnetic  meridian  for  that  place.  If 
the  magnet  is  mounted  on  a  horizontal  axis  so  that  it  can  turn 
freely  in  the  magnetic  meridian,  its  north  pole,  in  the  northern 
hemisphere,  will  generally  point  downward,  making  a  certain 
angle  with  the  horizontal  wrhich  is  called  the  magnetic  dip  for 
the  place.  In  the  southern  hemisphere  the  north  pole  points 
above  the  horizon.  By  magnetizing  a  globe  of  iron  and  exam- 
ining the  behavior  of  small  magnets,  suspended  freely  at  differ- 
ent points  on  its  surface,  Gilbert  reproduced  on  a  small  scale 
the  phenomena  exhibited  by  free  magnets  with  respect  to  the 
earth.  He  thus  concluded  that  the  earth  itself  is  a  magnet,  or 
at  least  has  magnetic  poles  like  those  of  a  magnet,  the  south 
magnetic  pole  of  the  earth,  toward  which  the  north  pole  of  the 
suspended  magnet  points,  being  situated  in  the  northern  hemi- 
sphere. 

This  conclusion  of  Gilbert's  is  in  general  correct,  but  there 
is  still  much  that  is  not  understood  about  the  magnetic  condi- 


MAGNETISM.  253 

tion  of  the  earth.  The  magnetic  meridians  do  not  coincide  with 
the  meridians  of  longitude,  nor  do  the  lines  of  equal  dip  coin- 
cide with  the  parallels  of  latitude.  The  magnetic  equator,  at 
which  there  is  no  dip,  is  an  irregular  line  crossing  the  geo- 
graphical equator  in  two  points.  The  angle  between  the  mag- 
netic meridian  and  the  geographical  meridian  at  any  place  is 
called  the  magnetic  variation  or  declination  at  that  place.  This 
variation  not  only  differs  in  different  places,  according  to  no 
well  defined  law,  but  it  also  changes  with  lapse  of  time.  It 
never  exceeds  a  certain  limit,  however,  and  appears  after  many 
years  to  go  through  a  cycle  of  values.  The  variation  is  also 
subject  to  a  small  diurnal  change.  The  dip  at  any  place  is  also 
subject  to  similar  changes. 

159.  Law  of  Magnetic  Force. — Hitherto  we  have  described 
in  general  terms  the  way  in  which  one  magnet  acts  on  another. 
In  order  to  construct  instruments  for  use  in  magnetic  measure- 
ments, and  in  order  to  advance  the  theory  of  magnetism,  experi- 
ments were  carried  out  by  Coulomb  to  determine  the  law  of  the 
force  between  two  magnetic  poles;  that  is,  to  determine  the 
way  in  which  the  force  between  the  poles  depends  on  the  poles 
themselves  and  on  the  distance  between  them.  In  these  experi- 
ments Coulomb  used  the  instrument  called  the  torsion  balance. 
This  instrument  depends  on  the  use  of  a  twisted  wire  for  meas- 
uring the  couple  by  which  it  is  twisted,  and  is  similar  in  all 
essential  features  to  that  used  by  Cavendish  in  determining  the 
gravitation  constant. 

With  this  torsion  balance  Coulomb  proved,  first,  that  the 
magnetic  forces  exerted  upon  the  magnet  by  the  earth  are 
always  applied  at  the  same  two  points  in  the  magnet,  which 
points  are  the  north  and  south  poles.  The  line  joining  these 
points  is  called  the  magnetic  axis,  and  when  the  magnet  is  free 
to  turn,  the  magnetic  axis  indicates  the  magnetic  north  and 
south  line.  The  forces  acting  on  these  poles,  because  of  the 
earth's  magnetic  condition,  are  equal,  parallel,  and  oppositely 
directed,  so  that  when  the  magnet  is  not  in  the  magnetic  merid- 
ian, it  is  acted  on  by  a  couple  which  tends  to  turn  it  into  that 
meridian.  In  observations  of  the  action  of  one  magnet  on 


254  MAGNETISM. 

another,  this  couple  due  to  the  earth  must  be  determined  and 
allowed  for.  When  Coulomb  determined  the  force  exerted  be- 
tween two  magnet  poles  at  different  distances  from  each  other, 
he  found  that  these  forces  were  to  each  other  inversely  as  the 
squares  of  the  respective  distances  between  the  poles.  This 
conclusion  that  the  force  between  two  magnet  poles  varies  in- 
versely with  the  square  of  the  distance  between  them,  was 
afterwards  confirmed  by  Gauss  in  another  way.  Gauss  assumed 
this  law  of  inverse  squares,  and  calculated  by  means  of  it  the 
forces  which  one  magnet  will  exert  upon  another  when  they  are 
placed  in  certain  different  positions  relative  to  each  other.  He 
then  examined  by  experiment  the  actual  forces  exerted  in  these 
different  positions,  and  found  them  to  be  in  accord  with  his 
calculations.  From  this  result  he  inferred  the  truth  of  the  law 
upon  which  the  calculations  were  based. 

160.  Quantity  of  Magnetism.  Strength  of  Pole. — Coulomb's 
law  of  the  force  between  two  magnetic  poles  recalls  the  law  of 
gravitation,  which  also  varies  inversely  with  the  square  of  the 
distance.  The  law  of  gravitation,  however,  contains  another 
statement,  namely,  that  the  force  is  proportional  to  the  inter- 
acting masses.  It  is  plain  that  in  some  way  the  force  between 
two  magnetic  poles  depends  on  what  we  may  call  the  strengths 
of  those  poles,  and  we  must  now  inquire  how  the  strength  of  a 
pole  can  be  measured,  and  how  the  force  between  two  poles 
depends  on  their  strength. 

In  the  case  of  gravitation,  the  masses  which  attract  each 
other  are  already  measured  in  terms  of  a  unit  mass,  and,  in 
establishing  the  law,  a  direct  proof  was  given  that  the  attrac- 
tion between  two  masses  is  proportional  to  them  both.  In  the 
case  of  magnetism,  we  have  no  such  independent  standard  of 
the  strength  of  a  pole,  or  of  what  we  may  call,  by  analogy,  the 
quantity  of  magnetism  at  the  pole.  We  have  no  evidence  that 
there  is  such  a  thing  as  magnetism.  We  only  know  that  a 
piece  of  steel  is  in  a  condition  in  which  it  exerts  peculiar  forces 
on  other  similar  pieces  of  steel,  and  we  have  no  way  of  meas- 
uring that  condition  except  by  those  forces.  We  are  therefore 
foiced  to  measure  the  strength  of  a  pole  by  the  force  which  it 


MAGNKT1SM.  265 

will  exert  on  a  standard  pole  according  to  some  assumed  law. 
The  law  which  we  assume  is  the  simplest  possible,  namely, 
that  the  strengths  of  different  poles  are  to  each  other  as  the 
forces  which  they  will  exert  on  the  same  pole,  if  placed  at  equal 
distances  from  it.  When  the  strength  of  pole  is  thus  defined, 
the  law  of  magnetic  force  may  be  enlarged  into  the  statement 
that  the  force  between  two  poles  is  proportional  to  the  product 
of  their  strengths  and  inversely  proportional -to  the  square  of 
the  distance  between  them.  Owing  to  the  fact  that  the  force 
between  two  poles  depends  on  the  medium  which  surrounds 
them,  we  assume  in  this  definition  that  the  poles  are  in  a 
vacuum.  The  force  measured  in  air  is  not  perceptibly  different 
from  that  in  vacuum. 

Our  definition  of  strength  of  pole  has  merely  compared  one 
pole  with  another.  It  is,  however,  important,  and  even  neces- 
sary for  magnetic  measurements,  to  adopt  a  unit  magnetic 
pole,  in  terms  of  which  all  other  poles  may  be  measured.  The 
way  in  which  such  a  pole  can  be  defined  is  best  seen  by  exam- 
ining the  formula  which  expresses  the  force  between  two  poles. 
By  introducing  a  factor  of  proportion  we  may  write  the  equa- 


poles  m  and  m',  when  the  distance  between  them  is  r.  The  sym- 
bol k  is  the  factor  of  proportion.  Now,  in  the  case  of  the  sim- 
ilar formula  expressing  the  law  of  gravitation,  the  two  masses, 
which  correspond  to  the  two  strengths  of  pole,  are  supposed 
to  be  known  in  terms  of  the  unit  of  mass,  and  the  force  and 
the  distance  are  measured,  so  that  by  the  substitution  of  these 
known  quantities  in  the  formula,  the  constant  k  is  determined 
as  the  gravitation  constant.  In  the  case  of  magnetism,  it  is 
conceivable  that  an  arbitrarily  chosen  magnetic  pole  might  be 
taken  as  a  standard,  and  other  magnetic  poles  always  ex- 
pressed in  terms  of  it.  If  this  were  done,  the  quantities  in  the 
formula  would  all  be  known,  except  the  factor  k,  and  so  fc 
would  be  determined  as  the  constant  of  magnetic  force.  It  is, 
however,  practically  impossible  to  construct  a  magnet  of  such 
a  sort  that  its  pole  can  conveniently  be  used  as  a  standard,  and 


256  MAGNETISM. 

it  is  equally  impossible  to  preserve  it  so  that  the  strength  of 
its  poles  will  not  change  with  use  and  with  lapse  of  time.  We 
therefore  proceed  otherwise  in  determining  the  unit  pole.  In- 
stead of  arbitrarily  choosing  a  unit  pole  and  then,  by  observa- 
tion of  the  force  between  two  measured  poles,  determining  the 
value  of  the  factor  of  proportion,  we  arbitrarily  choose  a  value 
of  that  factor,  and  then,  by  observation  of  the  force  between 
two  poles,  determine  their  values,  or  rather  the  value  of  their 
product.  To  make  this  choice  as  simple  as  possible,  we  set 
k=l.  On  this  assumption,  the  measured  force  between  two 
poles,  at  a  known  distance  apart,  will  enable  us  to  determine 
the  product  of  their  strengths.  If  the  poles  are  alike,  we  may 
thus  determine  the  strength  of  either  one  of  them.  In  partic- 
ular we  may  define  the  unit  pole,  or  pole  of  unit  strength,  as 
that  pole  which  will  repel  another  equal  and  similar  pole  at 
unit  distance  with  unit  force.  In  the  C.  G.  S.  system,  the  unit 
magnetic  pole  is  one  which  will  repel  an  equal  and  similar 
pole,  at  the  distance  of  one  centimetre,  with  the  force  of  one 
dyne. 

161.  Magnetic  Field. — When  a  small  magnet  is  brought 
near  another  magnet,  it  "will  be  acted  on  by  magnetic  forces, 
and  will  indicate  this  action  by  assuming  some  definite  posi- 
tion. If  we  imagine  it  possible  to  isolate  one  of  the  poles  of 
this  magnet,  or  what  amounts  to  the  same  thing,  if  we  exam- 
ine the  force  exerted  on  it,  we  shall  find  that,  in  these  circum- 
stances, it  is  acted  on  by  a  force  of  definite  amount  and  direc- 
tion. By  using  such  an  isolated  pole,  the  forces  in  the  region 
around  a  magnet,  or  around  any  combination  of  magnets,  can 
be  mapped  out  with  their  magnitudes  and  directions.  Such  a 
region,  in  which  a  magnetic  pole  will  be  acted  on  by  magnetic 
force,  is  called  a  magnetic  field. 

The  existence  of  a  magnetic  field  may  often  be  detected 
when  there  is  no' evidence  of  the  presence  of  a  magnet  to  which 
the  field  can  be  assigned.  Thus  there  is  a  magnetic  field 
around  a  wire  carrying  an  electric  current.  The  magnetic  field 
of  the  earth  is  another  example. 


MAONBTISM.  25.7 

if  we  lay  >a  sheet  of  stiff  paper  over  a  strong  bar  magnet, 
and  scatter  iron  filings  over  it,  the  filings  will  be  arranged  in 
curves  which  end  at  the  two  poles.  These  curves  mark  what 
Faraday  called  the  lines  of  force  of  the  magnet.  Described 
geometrically,  a  line  of  force  is  a  .line  in  the  field  of  force  so 
drawn  that  the  tangent  to  it  at  any  point  lies  in  the  direction 
of  the  force  at  that  point.  The  iron  filings,  by  which  the  lines 
of  force  around  a  magnet  are  marked  out,  are  generally  con- 
siderably longer  in  one  direction  than  in  others.  When  they 
fall  on  the  paper  they  are  magnetized  by  induction,  the  mag' 
netic  axis  in  each  one  generally  coinciding  with  its  greatest 
length.  The  little  magnets  thus  formed  point  along  the  lines 
of  force,  and  as  they  cling  together  by  their  mutual  attraction, 
they  mark  out  some  of  the  lines  of  force. 

Faraday  conceived -the  lines  of  force  around  a  magnet  to 
indicate  a  certain  condition  in  the  field,  and  to  arise  from  cer- 
tain conditions  in  the  body  of  the  magnet.  He  was  therefore 
led  naturally  to  consider  them  as  limited  in  number,  and  to 
conceive  that  a  certain  definite  number  of  them  proceeded  from 
a  pole  of  a  certain  strength,  or  that  the  strength  of  the  pole 
and  the  number  of  the  lines  of  force  which  proceeded  from  it 
were  in  proportion.  Without  adopting  the  physical  hypothesis 
of  Faraday,  we  may  use  the  conception  of  a  limited  number  of 
lines  of  force  to  describe  the  magnetic  field  with  respect  to  the 
force  which  is  exerted  at  any  point  in  it  upon  a  unit  pole. 
This  force  at  any  point  is  called  the  strength  of  the  field,  or  the 
magnetic  intensity  at  that  point.  Let  us  suppose  that  a  certain 
definite  number  of  lines  of  force  is  associated  with  each  mag- 
netic pole  of  unit  strength.  The  strength  of  any  pole  will  then 
be  represented  by  the  number  of  lines  of  force  associated  with 
it.  Now  in  drawing  the  diagram  of  lines  of  force,  we  proceed 
as  follows:  We  draw  a  surface  in  the  field  at  right  angles  to 
the  lines  of  force,  and  we  draw  the  lines  of  force  through  that 
surface,  so  distributed  that  the  number  of  lines  which  pass 
through  each  unit  of  area  of  the  surface  is  proportional  to  the 
strength  of  field  at  the  point  where  that  unit  area  is  situated. 
The  diagram  is  then  completed  by  continuing  these  lines  of 


268  MAGNETISM. 

force  to  their  ends.  With  this  diagram  the  strength  of  field 
anywhere  in  the  field  may  be  determined.  For  it  may  be  shown 
that  the  strength  of  field,  at  any  point  in  the  field,  is  then  pro- 
portional to  the  number  of  lines  of  force  which  will  pass 
through  a  unit  area,  set  up  at  that  point  perpendicularly  to 
the  lines  of  force. 

When  a  piece  of  iron  is  placed  in  a  matrnc-  ic  field,  it  be- 
comes magnetized  by  induction.  The  intensity  of  its  magnet- 
ization depends  on  the  strength  of  the  field,  and  the  direction 
of  the  axis  of  the  magnet  formed  by  induction  depends  on  the 
direction  of  the  lines  of  force  of  the  fieid.  We  may  state  the 
general  rule  for  the  direction  of  the  axis,  and  the  position  of 
the  poles,  by  saying,  that  the  positive  direction  of  tne  axis 
coincides  with  the  positive  direction  of  the  lines  of  force  of  the 
field.  By  the  positive  direction  of  the  axis,  we  mean  the  direc- 
tion from  the  south  to  the  north  pole  in  the  iron.  This  is  the 
direction  which  is  taken  as  positive  when  we  speak  of  the  point- 
ing of  a  compass  needle.  By  the  positive  direction  of  the  line 
of  force,  we  mean  the  direction  in  which  a  small  magnet  will 
point  if  placed  on  that  line. 

A  great  deal  of  attention  has  been  devoted  to  the  study  of 
the  magnetic  field  of  the  earth,  partly  for  scientific  reasons, 
but  mainly  on  account  of  its  great  importance  in  navigation 
and  surveying.  The  elements  of  the  magnetic  field  which  are 
examined  by  experiment  are  the  variation  or  declination,  the 
dip,  and  the  horizontal  intensity,  that  is,  the  horizontal  com- 
ponent of  the  strength  of  field.  As  to  the  method  of  determin- 
ing the  horizontal  intensity,  this  much  may  be  said.  If  a 
small  magnet  is  suspended  so  that  it  can  swing  freely  in  the 
horizontal  plane,  and  is  then  turned  out  of  the  magnetic  merid- 
ian, it  will  be  acted  on  by  a  couple  tending  to  bring  it  into  the 
magnetic  meridian.  For  a  swing  of  only  a  few  degrees,  the 
moment  of  this  couple  is  proportional  to  the  angular  devia- 
tion from  the  magnetic  meridian.  The  magnet  therefore  exe- 
cutes oscillations  which  are  similar  to  those  of  a  pendulum, 
and  the  time  of  oscillation  is  expressed  by  a  formula  like  the 
pendulum  formula,  in  which,  however,  the  moving  force  is  not 


MAGNBTISM.  259 

the  weight  of  the  body,  but  is  due  to  the  forces  exerted  by  the 
field  upon  the  poles  of  the  magnet.  By  determining  the  times 
of  oscillation  of  such  a  magnet  at  different  places  on  the  earth's 
surface,  the  horizontal  intensities  at  those  places  can  be  com- 
pared. 

The  value  of  such  determinations  as  these  depends  on  the 
constancy  of  the  magnet  with  which  they  are  made.  By  using 
the  same  magnet  to  deflect  another  one  from  the  magnetic 
meridian,  the  strength  of  its  poles  and  the  horizontal  intensity 
are  brought  into  another  relation,  and  by  a  combination  of 
this  relation  with  the  one  obtained  from  the  former  experi- 
ments, a  measure  of  the  horizontal  intensity  can  be  obtained 
in  absolute  units,  and  independent  of  the  particular  magnet 
with  which  the  experiments  are  made. 

It  is  found  that  the  horizontal  intensity,  like  the  variation 
and  the  dip,  undergoes  both  secular  and  daily  changes.  Not- 
withstanding the  great  amount  of  study  which  has  been  devoted 
to  the  earth's  magnetism,  no  satisfactory  theory  of  its  origin 
has  yet  been  given. 

162.  Para  magnetism  and  Diamagnetism. — Hitherto  we  have 
described  and  studied  magnetism  as  if  it  were  a  property  only 
of  iron.  This,  however,  is  not  the  case.  Iron  excels  all  other 
bodies,  in  a  very  marked  degree,  in  the  extent  to  which  it  ex- 
hibits magnetic  properties,  and  it  is  apparently  the  only  sub- 
stance from  which  permanent  magnets  can  be  made.  The  other 
metals  of  the  iron  group,  and  especially  nickel,  are  also  capable 
of  being  magnetized  by  induction  in  the  way  in  which  iron  is. 
They  are  called  paramagnetic  bodies.  A  few  other  substances, 
among  them  oxygen,  either  in  the  gaseous  state  or  in  the  liquid 
state,  are  paramagnetic. 

Until  1846  it  was  supposed  that  all  other  substances  were 
not  affected  in  the  least  in  a  magnetic  field,  but  in  that  year 
Faraday,  by  the  use  of  a  very  powerful  electromagnet,  found 
that  all  substances  upon  which  he  experimented  were  more  or 
less  affected  by  the  magnetic  field,  but  in  a  different  way  from 
that  in  which  the  paramagnetic  bodies  are  affected.  The  effect 
discovered  by  him  is  exhibited  most  strikingly  by  bismuth. 


260  MAOSKTISM, 

The  contrast  between  the  behavior  of  bismuth  and  of  iron  is 
shown  by  the  following  experiments:  If  a  rod  of  iron  is  sus 
pended  between  the  poles  of  a  strong  magnet,  it  will  turn  and 
point  along  the  lines  of  force.  The  action  is  in  accordance 
with  the  general  law  of  induction,  which  has  already  been 
stated.  On  the  other  hand,  if  a  rod  of  bismuth  is  suspended  in 
the  same  field,  it  will  turn  until  it  points  across  the  lines  of 
force.  Since  the  bismuth  does  not  retain  any  traces  of  magnet- 
ization when  it  is  removed  from  the  field,  we  are  not  able  to 
tell  exactly  what  condition  it  assumes  when  in  the  field,  but 
its  behavior  in  the  field  can  be  explained  by  supposing  that  it 
becomes  a  magnet,  so  developed  that  the  positive  direction  of 
its  axis  is  opposite  to  the  positive  direction  of  the  lines  of  force 
of  the  field.  Bodies  which  exhibit  this  property  are  called 
diamagnetic  bodies.  The  action,  even  in  the  most  pronounced 
cases,  is  extremely  feeble. 

The  hypothesis  made  in  the  last  paragraph,  that  the  mag- 
netic arrangement  in  a  diamagnetic  body  is  opposite  to  that 
which  is  set  up  in  a  paramagnetic  body,  is  extremely  difficult 
to  reconcile  with  our  theories  of  magnetism.  An  experiment 
tried  by  Faraday  indicates  a  way  by  which  that  difficulty  may 
be  avoided.  Faraday  filled  a  small  glass  tube  with  a  solution 
of  sulphate  of  iron,  which  is  a  paramagnetic  body.  When  this 
tube  was  suspended  in  the  magnetic  field,  it  pointed  along  the 
lines  of  force.  When,  however,  it  was  immersed  in  a  stronger 
solution  of  sulphate  of  iron,  it  pointed  across  the  lines  of  force. 
This  experiment  indicates  the  following  hypothesis  to  account 
for  the  contrast  between  paramagnetic  ana  diamagnetic  bodies ; 
that  the  ether  is  itself  a  magnetizable  body,  and  that  para- 
magnetic bodies  acquire  a  greater  intensity  of  magnetization 
than  the  ether  around  them,  while  diamagnetic  bodies  acquire 
a  less  intensity  of  magnetization.  There  is  no  way  by  which 
these  hypotheses  can  be  tested  by  experiment. 

163.  Causes  Affecting  Magnetisation. — We  have  used  the 
phrase,  intensity  of  magnetization,  to  express  the  magnetic 
condition  of  a  body.  This  phrase  must  now  be  defined.  In 
order  to  do  so,  we  first  define  the  magnetic  moment  01  a  mag- 


MAGNETISM.  261 

net,  whether  permanent  or  induced,  as  the  product  of  the 
strength  of  one  of  its  poles  and  the  distance  between  its  poles. 
Its  intensity  of  magnetization  is  then  defined  as  the  quotient 
of  its  magnetic  moment,  divided  by  its  volume,  or  the  magnetic 
moment  of  unit  of  volume.  It  will  easily  be  seen  that,  when 
the  unit  of  volume  is  taken  small  enough,  this  quantity  char- 
acterizes the  magnetic  condition  of  each  part  of  the  magnet. 
The  definition  assumes  that  the  magnetism  exhibited  by  the 
magnet  is  due  to  a  condition  which  exists  in  all  parts  of  it. 
This  assumption  is  suggested  by  the  experiment  of  Gilbert 
already  referred  to,  and  is  justified  by  the  facts  which  are  now 
to  be  cited. 

When  a  piece  of  soft  iron  is  brought  into  a  magnetic  field, 
it  is  immediately  magnetized  by  induction.  Its  intensity  of 
magnetization  does  not  reach  its  full  value  instantly.  If  the 
strength  of  the  field  is  gradually  increased,  the  intensity  of 
magnetization  also  increases.  The  two  quantities  at  first  in 
crease  proportionally,  but  as  the  strength  of  the  field  increases, 
the  intensity  of  magnetization  for  a  time  increases  faster  and 
then  more  slowly  than  the  strength  of  the  field,  and  at  last 
reaches,  or  nearly  reaches,  a  limit.  The  iron  is  then  said  to  be 
saturated.  If  the  strength  of  field  is  now  diminished  until  it 
gradually  becomes  zero,  the  intensity  of  magnetization  dimin- 
ishes, but  does  not  entirely  vanish.  In  order  to  reduce  it  to 
zero,  the  field  must  be  reversed,  and  its  strength  in  the  reverse 
direction  raised  to  a  certain  value.  After  it  has  passed  this 
value,  the  intensity  of  magnetization  also  reverses,  and  in- 
creases in  the  reverse  direction  until  the  iron  is  again  satu- 
rated. By  a  repetition  of  this  process,  the  magnetization  of 
the  iron  can  be  carried  through  a  cycle  of  values.  For  each 
value  of  the  strength  of  field  there  will  be  two  values  of  the 
intensity  of  magnetization,  depending  on  whether  the  particu- 
lar strength  of  field  has  been  reached  by  increasing,  or  by 
decreasing,  the  field  strength.  This  general  behavior  of  iron, 
by  which  its  magnetization  seems  to  lag  behind  the  magnet- 
izing force,  is  called  by  Ewing,  who  investigated  it,  hysteresis. 


262  MACINKTISM. 

If  a  piece  of  steel  is  placed  in  a  magnetic  field,  it  will  also 
become  a  magnet,  though  not  so  rapidly,  nor  to  such  an  extent, 
as  an  equal  piece  of  iron.  Its  magnetization  can  be  enhanced 
by  striking  it  so  as  to  make  it  ring.  When  it  is  removed  from 
the  magnetic  field,  it  does  not  lose  its  magnetism,  when  struck 
or  jarred,  as  the  iron  does,  but  retains  at  least  the  greater  part 
of  it.  It  is  thus  a  permanent  magnet.  By  striking  it  again, 
its  magnetization  will  be  diminished.  If  it  is  heated,  its  mag- 
netization will  also  diminish,  but  will  return  to  its  original 
value,  or  nearly  so,  when  the  magnet  is  cooled,  unless  the  heat- 
ing has  been  carried  beyond  a  certain  limit.  If  the  tempera- 
ture of  the  magnet  is  carried  above  785°  C.,  the  magnetization 
entirely  disappears,  and  is  not  restored  on  cooling.  A  piece  of 
iron  raised  above  that  temperature  and  brought  into  a  mag- 
netic field  exhibits  no  magnetic  properties  whatever,  until  its 
temperature  falls  below  that  temperature. 

164.  Theories  of  Magnetism. — When  the  actions  of  magnets 
were  first  studied,  they  were  ascribed  to  the  presence  in  the 
magnet  of  certain  effluvia,  or  magnetic  fluids,  which  were 
assumed  to  be  of  two  opposite  kinds,  possessing  the  properties 
of  repelling  fluid  of  the  same  kind  and  of  attracting  fluid  of 
the  opposite  kind.  These  fluids  were  supposed  to  be  present 
in  equal  quantities  in  any  mass  of  iron,  and  to  be  separated  by 
the  process  of  induction.  The  fluid  collected  in  one  end  of  the 
magnet  formed  the  north  pole,  and  that  in  the  other  end,  the 
south  pole.  Gilbert's  discovery  that  when  a  magnet  wag 
broken,  the  two  parts  did  not  contain  simply  a  north  and  a 
south  pole  respectively,  but  became  complete  magnets,  made  it 
necessary  to  suppose  that  these  fluids  existed  in  each  particle 
of  iron,  and  were  separated  in  it  by  induction,  so  that  each 
particle  in  the  magnetized  iron  became  a  magnet.  The  theory 
of  magnetism  was  developed  in  this  form  by  Poisson,  and 
showed  itself  capable  of  accounting  for  the  laws  oi  magnetic 
force,  the  phenomena  of  magnetic  induction,  and  of  the  dis- 
tribution of  magnetic  force  in  a  magnet. 

This  theory  does  not  explain  so  readily  the  facts  which  have 
been  described  in  the  last  section.  In  order  to  explain  them, 


MAGNETISM.  263 

Weber  proposed  another  theory,  which,  as  modified  by  Ewing, 
is  the  one  now  generally  accepted.  It  does  not  attempt  to 
account  for  magnetism  itself,  but  only  to  explain  the  behavior 
of  magnetized  bodies.  It  is  rather  a  theory  of  the  structure  of 
a  magnet  than  a  theory  of  magnetism.  Weber  supposed  each 
molecule  of  iron  to  be  a  magnet.  In  the  unmagnetized  condi- 
tion of  the  iron  these  molecular  magnets  have  no  definite  ar- 
rangement. The  magnetization  of  the  iron  consists  in  arrang- 
ing the  molecular  magnets  in  chains  or  rows.  If  we  consider 
the  magnetic  forces  which  a  single  row  of  molecular  magnets 
thus  arranged  will  exhibit,  it  appears  that  the  contiguous  poles 
of  two  neighboring  molecules  will  mutually  destroy  each  other's 
effects,  so  that  the  only  poles  which  will  be  effective  in  produc- 
ing an  external  magnetic  field  will  be  the  north  pole  of  a 
molecule  at  one  end  of  the  row  and  the  south  pole  of  a  molecule 
at  the  other  end.  If  we  consider  a  magnet  to  be  a  collection  of 
such  rows  laid  side  by  side,  and  take  into  account  the  fact  that 
the  mutual  repulsion  of  the  similar  free  poles  at  the  ends  of 
the  rows  will  tend  to  make  those  ends  diverge  from  each  other, 
we  can  account  for  the  existence  of  magnetic  poles,  for  the  dis- 
tribution of  magnetic  force  over  the  ends  of  a  magnetized  bar, 
and  for  the  fact  that  the  intensity  of  magnetization  of  a  mag- 
net is  greater  in  the  middle  than  it  is  at  the  ends.  The  effect 
produced  by  striking  a  magnet  is  also  explained  by  this  theory ; 
for,  we  may  suppose  that  any  one  molecule  of  the  iron  is 
restrained  somewhat  from  pointing  in  the  direction  of  the  lines 
of  force  of  the  field  by  the  mechanical  obstruction  of  the  mole- 
cules around  it,  and  that  if  it  is  momentarily  freed  from  this 
obstruction  by  the  jar  produced  by  striking  the  magnet,  it  will 
swing  into  position  more  readily ;  or  if  it  is  already  in  position, 
and  the  magnet  is  not  in  a  magnetic  field,  will  swing  more  or 
less  out  of  position,  so  as  partly  to  disintegrate  the  rows.  The 
effect  produced  by  heating  the  magnet,  which  gives  its  mole- 
cules more  freedom  of  motion,  is  explained  in  a  similar  manner 
To  explain  hysteresis  we  must  consider,  with  Ewing,  that 
the  arrangement  of  an  assemblage  of  molecular  magnets  is  not 
an  entirely  unordered  or  irregular  one,  but  that  by  their 


264  MAGNETISM. 

mutual  attractions  the  molecules  will  arrange  themselves  in 
••stable  groups.  When  a  magnetizing  force  is  applied  to  a  col- 
lection of  such  groups,  the  first  effect  will  be  to  turn  the  mole 
cules  of  the  group  toward  the  line  of  the  force,  and  so  to 
increase  the  intensity  of  magnetization.  When  the  magnetizing 
force  reaches  a  certain  limit,  the  groups,  one  by  one,  cease  to 
be  stable.  The  molecules  in  them  then  turn  freely  under  the 
magnetizing  force,  and  form  new  stable  groups,  of  which  the 
intensity  of  magnetization  is  much  greater.  This  description 
is  in  accord  with  the  experimental  fact  that  the  magnetization 
of  iron  undergoes  a  great  increase  for  a  comparatively  small 
change  of  the  magnetizing  force,  between  certain  limiting 
values.  After  these  new  groups  are  formed,  the  further 
increase  of  the  magnetizing  force  merely  brings  the  molecules 
of  the  groups  more  nearly  parallel  to  each  other.  Complete 
saturation  will  be  reached  when  they  are  all  arranged  in 
parallel  rows.  When  the  magnetizing  force  is  again  dimin- 
ished, the  stability  of  the  groups  that  have  been  formed  con- 
tinues even  after  the  value  of  the  magnetizing  force  is  less  than 
that  at  which  the  groups  were  formed,  and  the  groups  do  not 
break  down  and  assume  their  original  condition  until  the 
magnetizing  force  is  considerably  diminished. 

It  is  evident  that  this  theory  of  Ewing  explains  all  the 
facts  which  we  have  explained  by  the  theory  of  Weber. 

The  subject  of  magnetism  is  intimately  connected  with  that 
of  the  electric  current,  and  will  be  considered  again  when  we 
deal  with  the  properties  of  the  current. 


STATIC   BLECTBICITT.  265 


STATIC   ELECTRICITY. 

165.  Electric  Attraction. — If  a  piece  of  amber  is  gently 
rubbed  on  a  dry  cloth,  and  is  then  brought  near  small  bits  of 
straw  or  paper,  these  light  bodies  will  be  attracted  by  it,  and 
will  adhere  to  it.    This  observation  is  a  very  ancient  one.     It 
is  said  to  have  been  known  to  Thales    (600  B.  C.).     By  the 
ancient  as  well  as  by  the  mediaeval  philosophers,  this  action 
was  looked  on  as  something  occult,  and  was  supposed  to  be 
peculiar  to  amber,  just  as  the  attraction  of  iron  was  peculiar 
to  the  lodestone.     It  was  not  investigated  in  a  scientific  way 
until  it  was  studied  by  Gilbert,  in  connection  with  his  study 
of  the  magnet.     Gilbert  mounted  a  light  pointer  on  a  needle, 
so  that  it  could  turn  about  freely,  and  with  it  as  an  indicatoi 
he  examined  other  bodies  besides  amber,  to  see  whether  they 
would  exhibit  a  similar  attractive  power.     Sulphur  had  appar- 
ently  been   found  by   some   previous  observer   to  behave   like 
amber,  and  Gilbert  found  that  many  other  bodies  behaved  in 
a  similar  way.     From  the  Greek  name  for  amber,  electron,  he 
called  this  action  electric  action.     We  now  use  the  word  elec- 
tricity to  express  the  assumed  cause  of  this  action.     A  body 
exhibiting  electric  action   is   said  to   be   electrified,  or  to   be 
charged  with  electricity. 

Von  Guericke  made  an  important  observation  by  noticing 
that  the  light  bodies,  which  were  attracted  to  the  electrified 
body,  were  repelled  by  it  after  a  time. 

166.  Electric    Conductors. — In    experimenting   on   electric 
action,  Grey  discovered  that  when  certain  bodies  were  brought 
in   contact   with   an   electrified  body,  they   also   acquired   the 
power  of  attracting  light  bodies.    He  next  undertook  to  trans 
mit   this   power    from   the   electrified   body    to    another    body 
through  a  long  thread.     In  his  first  experiment,  he  hung  an 
ivory  ball  by  a  thread  from  an  upper  window,  and  found  that 
when  the  electrified  body  was  touched  to  the  upper  end  of  the 
thread,  the  ball  became  electrified.     He  tried  next  a  similar 


260  STATIC    ELECTRICITY. 

experiment,  in  which  the  thread  was  suspended  horizontally 
by  loops  of  thread,  and  found  that  in  this  case  the  ball  was  not 
electrified.  Considering  the  reason  of  his  success  in  the  one 
case  and  his  failure  in  the  other,  he  conceived  that  it  might  be 
due  to  the  conducting  away  of  the  electricity  by  the  loops  on 
which  the  thread  was  suspended  in  the  second  experiment.  He 
accordingly  suspended  the  thread  by  other  bodies,  and  found 
at  last  that,  when  silk  thread  for  the  suspension  was  used,  the 
action  was  transmitted  to  the  ball.  Grey  thus  proved  the  dis- 
tinction between  conductors  and  non-conductors.  Many  bodies, 
of  which  the  metals  are  the  most  conspicuous,  transmit  the 
electric  condition  from  one  body  to  another,  and  are  classed  as 
conductors.  Other  bodies,  of  which  glass,  the  resins,  and  silk 
are  examples,  transmit  this  condition  either  very  slowly,  or 
not  to  any  perceptible  degree,  and  are  classed  as  non- 
conductors. Subsequent  investigation  has  shown  that  there  is 
no  sharp  line  of  separation  between  these  two  classes  of  bodies. 
Probably  all  bodies  conduct  to  some  extent.  The  distinction,, 
however,  between  conductors  and  non-conductors,  is  generally 
made,  and  many  substances  are  so  nearly  non-conductors,  that 
they  can  be  treated  as  such  in  most  experiments. 

107.  Electric  Attractions  and  Repulsions. — Putting  to- 
gether the  transmission  of  the  electric  condition,  as  observed 
by  Grey,  and  the  repulsion  of  light  bodies  from  the  electrified 
amber  after  contact  with  it,  as  observed  by  Von  Guericke,  it 
was  easy  to  see  that  the  repulsion  might  be  due  to  the  similar 
electric  condition  of  the  bodies.  The  experiment  was  there- 
fore suggested  of  examining  the  way  in  which  various  electri- 
fied bodies  act  on  each  other.  Such  experiments  were  carried 
out  by  Dufay  in  1733,  and  a  few  years  later  by  Franklin. 
These  investigators  found  that  all  bodies  which  can  be  electri- 
fied by  friction  can  be  placed  in  one  of  two  classes,  according 
as  they  repel  or 'attract  an  electrified  piece  of  glass.  These 
experiments  are  most  easily  conducted  by  using  as  an  indi- 
cator a  gilded  pith  ball  hung  by  a  non-conducting  silk  thread. 
If  a  rod  of  glass  is  electrified  by  friction,  and  brought  near 
this  ball,,  it  will  at  first  attract  the  ball  to  it.  As  soon,  how- 


STATIC    KLECTRICITY.  267 

6ver,  as  the  ball  acquires  the  electric  condition  by  contact  with 
the  glass,  it  is  repelled  from  the  glass.  Its  electric  condition 
is  then  the  same  as  that  of  the  glass,  and  it  can  be  used  in- 
stead of  a  piece  of  glass  in  testing  other  bodies.  According  to 
Dufay,  it  is  vitreously  electrified,  or  charged  with  vitreous 
electricity.  According  to  Franklin,  who  adopted  another  the- 
ory and  considered  the  effects  to  be  described  as  due  to  the 
excess  or  defect  in  the  body  of  a  single  substance,  it  is  posi- 
tively electrified.  Franklin's  terms  are  those  now  generally 
used,  although  the  views  which  led  to  them  are  no  longer  held. 

To  test  the  electric  relations  of  various  bodies,  they  are 
submitted  to  friction,  and  then  presented  to  the  positively 
electrified  indicator.  Some  of  the  bodies  thus  treated  will 
repel  the  ball  as  the  glass  did,  and  are  therefore  in  the  same 
electric  condition  as  that  of  the  glass.  The  other  bodies  will 
attract  the  ball.  The  question  arises  whether  this  attraction 
is  similar  to  that  exerted  by  the  glass  on  the  ball  in  the  first 
instance,  or  whether  it  is  something  different.  To  test  this, 
the  ball  is  touched  with  the  hand,  so  that  its  electric  condition 
is  removed.  One  of  the  bodies  which  causes  attraction  of  the 
positively  electrified  ball  is  then  presented  to  it.  The  ball 
behaves  exactly  as  it  did  when  the  glass  was  brought  near  it; 
that  is,  it  is  first  attracted,  until  it  comes  in  contact  with  the 
body,  and  is  then  repelled.  It  is  therefore  presumably  in  the 
same  electric  condition  as  that  of  the  body  which  it  has 
touched.  In  this  condition  it  is  repelled  by  all  the  bodies 
which  attracted  it  when  it  was  positively  electrified,  and  is 
attracted  by  all  the  bodies  which  then  repelled  it.  The  ball 
has  therefore  acquired  another  condition,  in  which  the  forces 
which  it  exerts  are  opposite  to  those  which  it  exerts  when 
positively  electrified.  This  other  condition  is  developed  on 
resin,  and  Dufay  therefore  said  that  the  ball  in  this  condition 
is  resinously  electrified,  or  is  charged  with  resinous  electricity. 
In  accordance  with  his  theory.  Franklin  said  that  it  is  nega- 
tively electrified.  This  is  the  term  which  we  now  use  to  ex- 
press this  condition. 

Dufay  summed  up  the  results  of  his  investigation  in  the 
law  that  there  exist  two  electric  conditions,  such  that  two 


ZOO  STATIC  ELECTRICITY. 

bodies  in  the  same  condition  repel  each  other,  while  two  bodies 
in  different  conditions  attract  each  other.  This  law  may  be 
otherwise  stated  by  saying  that  similarly  electrified  bodies 
repel  each  other,  while  dissimilarly  electrified  bodies  attract 
each  other.  As  we  shall  subseqeuntly  find  that  the  super- 
position of  dissimilar  electricities  on  the  same  body  results  in 
the  disappearance  of  both  of  them,  we  may  also  speak  of  oppo- 
sitely, instead  of  dissimilarly,  electrified  bodies. 

168.  The  Electric    Spark. — Von   Guericke   constructed   an 
electric  machine  by  mounting  a  ball  of  sulphur  so  that  it  could 
be  turned  around  an  axis,  and  setting  near  it  a  metallic  body 
supported  by  a  non-conductor.     When  the  ball  was  turned,  and 
the  dry  hands,  or  a  piece  of  flannel,  were  pressed  upon  it,  it 
became  negatively  electrified,  and  imparted  negative  electricity 
to  the  metallic  conductor.     With  this  machine,  Von  Guericke 
found  that,  after  the  conductor  had  been  receiving  electricity 
for  some  time,  a  spark  would  pass  from  it  to  any  conducting 
body  held  near  it. 

Other  electric  machines  were  soon  constructed  which  were 
far  more  efficient  than  this  primitive  one,  and  with  them  much 
larger  sparks  were  obtained.  These  sparks  were  generally 
taken  to  be  evidence  of  the  passing  of  something,  called  elec- 
tricity, from  the  charged  body. 

169.  The  Leyden  Jar. — Cunaeus  of  Leyden  attempted  to 
collect  electricity  in  water  by  holding  a  glass  of  water  near  the 
conductor  of  an  electric  machine,  and  allowing  sparks  to  pass 
through  a  nail  partly  immersed  in  the  water.    After  the  sparks 
had  passed  for  some  time  he  attempted  to  withdraw  the  nail, 
and  in  so  doing  received  a  shock  through  his  arms  and  body. 
In  the  discovery  of  this  effect  he  had  been  anticipated  by  von 
Kleist,  but  Cunaeus'  account  of  his  discovery  attracted  atten- 
tion, and  the  instrument  by  which  the  shock  could  be  given 
was  called  the  Leyden  jar.    This  name  is  still  in  use,  although 
the  Leyden  jar  is  simply  one  example  of  the  class  of  instru- 
ments known  as  electric  condensers. 

In  its  present  form  the  Leyden  jar  is  a  glass  jar  coated 
within  and  without,  over  its  bottom  and  the  lower  part  of  its 


STATIC   ELECTRICITY;.  269 

sides,  with  tinfoil.  Through  the  stopper  in  the  neck  is  fixed  a 
metallic  rod,  ending  above  in  a  ball  or  knob,  and  connected 
with  the  inner  coating  of  the  jar  by  a  chain.  To  charge  the 
jar  the  outer  coating  is  touched  with  the  hand,  or  is  otherwise 
brought  in  conducting  contact  with  the  earth,  and  sparks  are 
allowed  to  pass  to  the  knob.  After  they  have  passed  for  some 
time,  the  jar  may  be  discharged  by  touching  one  end  of  a  con- 
ducting body  to  the  outer  coating,  and  bringing  tue  other  end 
of  it  near  the  knob.  A  spark  will  pass  between  the  knob  and 
the  'conductor,  which  is  brighter  and  thicker  and  makes  a 
louder  report  than  any  of  the  sparks  by  which  the  jar  was 
charged.  The  successive  charges  which  pass  to  the  jar  from 
the  machine  seem  to  have  been  -accumulated  or  condensed  in 
the  jar.  When  the  discharge  is  taken  through  the  body,  the 
shock  is  often  very  severe. 

•  The  explanation  of  the  action  of  the  Leyden  jar  was  first 
given  by  Franklin.  In  order  to  appreciate  it,  we  must  con- 
sider the  way  in  which  an  electrified  body  acts  on  bodies  near 
it,  to  produce  the  electric  condition  in  them. 

170.  Electric  Induction. — In  examining  the  effects  of  an 
electrified  body  on  other  bodies,  it  is  necessary  to  mount  these 
bodies  on  non-conductors,  or  to  insulate  them.  When  a  con- 
ductor thus  insulated  is  brought  near  an  electrified  body,  it 
will  exhibit  signs  of  electrification.  We  test  the  peculiarities 
of  its  electrification  by  the  use  of  a  proof-plane.  This  is  a 
small  disk  of  sheet  metal  mounted  on  a  non-conducting  handle. 
When  the  proof-plane  is  laid  on  the  surface  of  the  conductor 
at  any  point,  it  acquires  a  charge  similar  to  that  of  the  con- 
ductor at  that  point,  and  by  removing  it  and  testing  the 
charge  on  it,  the  charge  on  the  conductor  can  be  determined. 
By  the  use  of  such  a  proof-plane  it  is  shown  that  the  parts  of 
the  conductor  already  described  which  are  farthest  from  the 
electrified  body  have  a  charge  like  that  of  the  electrified  body, 
and  that  those  parts  which  are  nearest  the  electrified  body 
have  a  charge  of  the  opposite  kind.  These  charges  are  shown 
to  be  equal  in  amount  by  removing  the  conductor  from  the 
presence  of  the  electrified  body,  for  then  no  charge  can  be  de- 


270  STATIC  ELECTRICITY. 

tected  on  it.  The  action  here  described,  by  means  of  which  an 
electrified  body  can  excite  the  electric  condition  in  a  neighbor- 
ing body,  which  is  insulated  from  it,  is  called  electric  induc- 
tion. 

The  attraction  between  a  charged  body  and  any  other  body 
brought  near  it  is  due  to  the  attraction  between  the  original 
charge  of  the  first  body  and  the  neighboring  charge  of  the 
opposite  kind  induced  in  the  second  body.  When  the  second 
body  is  insulated  and  contact  occurs  between  the  bodies,  this 
charge  of  opposite  kind  is  destroyed  by  union  with  some  of  the 
original  charge  and  both  bodies  become  charged  similarly.  In 
this  condition  they  repel  each  other. 

If  a  conductor,  when  in  the  presence  of  a  charged  body  and 
charged  by  induction  with  both  kinds  of  electricity,  is  touched 
by  the  hand,  or  is  connected  with  the  earth  by  any  conductor, 
the  charge  which  it  has  on  the  end  farthest  from  the  electrified 
body,  and  which  is  of  the  same  sort  as  that  of  the  electrified 
body,  disappears.  The  other  charge,  on  the  end  nearest  the 
electrified  body,  remains  unchanged.  These  two  charges,  there- 
fore, seem  to  differ,  in  that  one  of  them  is  free  to  leave  the 
conductor  by  passing  through  any  conductor  presented  to  it, 
while  the  other  is  bound  or  retained  in  the  conductor.  They 
are  called  the  free  and  the  bound  charge  respectively.  The 
experiments  here  detailed  show  that  the  bound  and  the  free 
charges  are  equal  in  magnitude,  and  that  the  bound  charge  is 
the  one  which  is  opposite  in  kind  to  the  original  charge. 

To  apply  these  facts  to  the  explanation  of  the  behavior  of 
the  Leyden  jar,  we  notice  that,  when  the  jar  is  being  charged, 
the  inner  coating  receives  a  charge  directly  from  the  machine, 
and  acts  as  the  original  electrified  body.  By  the  action  of  the 
charge  which  it  first  receives,  the  outer  coating  is  charged  by 
induction,  and  its  induced  free  charge  passes  off  to  the  earth. 
Its  bound  charge  acts  by  induction  on  the  inner  coating,  so  as 
to  make  a  free  charge  on  that  coating  of  the  opposite  sort  to 
that  which  is  supplied  by  the  machine.  The  machine  therefore 
can  supply  an  additional  charge  to  neutralize  this,  and  the 
charge  on  the  inner  coating  is  increased  by  the  amount  of  the 


HTAT1C    KLECTR1CITT.  271 

bound  charge  which  was  developed  in  it.  This  increased 
charge  on  the  inner  coating  again  acts  by  induction,  to  increase 
the  bound  charge  on  the  outer  coating,  and  this  process  goes  on 
until  the  jar  discharges  itself  or  until  a  limit  is  reached, 
which  is  determined  by  the  nature  of  the  electric  machine  that 
is  supplying  the  charge.  In  the  final  condition  of  the  jar  when 
charged,  the  conductor  of  the  machine  and  the  knob  of  the  jar 
are  in  similar  electric  conditions,  so  that  no  farther  charge 
can  pass  between  them,  and  two  opposite  charges,  of  practi- 
cally equal  magnitude,  confront  each  other  on  the  surfaces  of 
the  two  coatings  which  are  next,  to  the  glass.  When  conduct- 
ing connection  is  made  between  the  outer  and  the  inner  coat- 
ing, these  charges  recombine  through  the  conductor,  and  the 
jar  is  discharged. 

The  laws  of  electric  induction  are  also  illustrated  in  an 
instrument  invented  by  Volta,  known  as  the  electrophorus.  It 
consists  of  a  sheet  of  sulphur,  rubber,  or  some  similar  sub- 
stance, which  can  be  electrified  by  friction,  called  the  plate, 
and  a  metallic  disk  furnished  with  an  insulating  handle, 
called  the  carrier.  The  plate  is  electrified  by  friction,  generally 
negatively,  and  the  carrier  placed  upon  it.  In  this  position 
the  carrier  is  usually  supported  on  slight  prominences  in  the 
plate,  so  that  it  is  in  contact  with  the  plate  at  very  few  points, 
and  does  not  receive  a  charge  from  it  by  conduction.  It  is 
charged  by  induction,  the  charge  on  its  lower  surface  being 
.positive,  opposite  to  that  of  the  plate.  The  negative  charge  on 
its  upper  surface  is  free,  and  is  removed  by  touching  the  car- 
rier with  the  hand.  When  the  carrier  is  lifted  from  the  plate, 
the  bound  positive  charge,  being  no  longer  constrained  by  the 
presence  of  the  negative  charge  of  the  plate,  distributes  itself 
over  the  carrier.  This  charge  may  then  be  communicated  to 
any  conductor  which  it  is  desired  to  charge,  and  the  carrier 
may  be  charged  again  in  a  similar  manner,  as  often  as  we 
please,  the  original  charge  on  the  plate  being  retained  by  it 
and  remaining  unaltered  by  the  operation. 

171.  Distribution  of  Electricity. — On*  the  hypothesis  that 
the  electric  charge  is  some  sort  of  fluid  whose  parts  repel  each 


272  STATIC   ELECTRICITY. 

other,  it  ,is  easy  to  see  that  the  distribution  of  electricity  in  an 
insulated  conductor  will  depend  upon  the  shape  of  the  con- 
ductor It  was  shown  first  by  Franklin,  and  long  afterwards 
more  conclusively  by  Faraday,  that  whatever  be  the  shape  of 
the  conductor,  the  charge  in  it  resides  entirely  on  its  surface. 
By  testing  different  bodies  with  the  proof-plane,  it  was  also 
shown  that  the  density  of  the  charge,  that  is,  the. amount  on 
unit  of  surface,  is  greatest  at  the  parts  of  greatest  curvature. 
When  the  law  of  electric  force  was  discovered,  the  distribution 
on  bodies  of  particular  forms  was  calculated.  The  experi- 
mental examination  of  the  distribution  on  such  bodies  verified 
the  results  of  these  calculations. 

The  density  of  the  charge  is  especially  great  at  any  part  of 
the  conductor  which  ends  in  a  sharp  point.  At  such  points, 
the  pressure  of  the  dense  charge  against  the  surrounding  insu- 
lator is  especially  great,,  and  the  charge  therefore  passes  from 
them  with  great  facility.  This  property  of  points,  of  dis- 
charging bodies  and  also  of  receiving  charges,  is  made  use  of 
in  all  sorts  of  electric  apparatus. 

.  172.  Early  Theories  cf  Electricity. — After  Dufay  had  dis- 
covered the  opposite  effects  produced  by  the  charges  on  glass 
and  on  resin,  he  developed  a  theory  of  electricity  to  explain 
the  facts.  He  supposed  that  there  exist  in  the  universe  two 
fluids,  which  he  called  vitreous  and  resinous  electricity;  that 
in  any  uncharged  body  these  fluids  exist  in  equal  amount ;  that 
they  are  separated  from  each  other  by  friction,  one  of  the  two 
bodies  which  are  rubbed  together  retaining  a  surplus  of  the 
one  fluid,  while  the  other  body  has  a  surplus  of  the  other  fluid ; 
that  the  parts  of  either  of  these  fluids  repel  other  parts  of  the 
same  fluid  and  attract  parts  of  the  other  fluid.  This  theory, 
which  is  commonly  known  as  the  two-fluid  theory  of  electricity, 
was  not  at  first  accepted,  but  it  gradually  won  its  way  into 
prominence,  and  is  the  one  which  has  determined  most  of  the 
nomenclature  of  electricity.  Franklin  adopted  the  view  that 
there  exists  in  the  universe  a  single  fluid,  called  electricity,  and 
that  a  body  when  charged  positively  possesses  a  surplus  of  this 
fluid,  while  a  body  charged  negatively  possesses  less  than  the  ' 


STATIC   ELECTRICITY.  273 

normal  amount.  This  theory  was  developed  by  Aepinus,  and 
for  some  time  was  generally  held.  It  fell  into  disuse  after  a 
while,  in  consequence  chiefly  of  the  necessity  which  Aepinus 
found,  in  order  to  explain  the  electric  attractions  and  repul- 
sions, as  well  as  the  passivity  of  unelectrified  bodies,  of  sup- 
posing that  matter  repelled  matter  according  to  the  same  law 
as  that  by  which  electricity  repelled  electricity.  Without  the 
assumption  of  a  mutual  repulsion  between  the  particles  of 
matter,  as  well  as  of  an  attraction  between  matter  and  elec- 
tricity, the  actual  forces  observed  could  not  be  explained. 

The  modern  theories  of  electricity  unite  in  assuming  that 
electricity,  whatever  it  may  be,  is  not  a  continuous  fluid,  but 
exists  in  separate  portions,  or  units,  which  may  be  called  elec- 
tric atoms  or  electrons. 

173.  Law  of  Electric  Force. — For  any  further  development 
of  the  subject,  it  becomes  important  to  know  the  law  of  the 
force  exerted  by  one  electric  charge  on  another.  The  law  was 
first  investigated  by  Cavendish  about  the  year  1773.  His 
method  of  investigation  depended  on  a  theorem  first  proved  by 
Newton,  to  the  effect  that  the  strength  of  field  at  any  point 
within  a  uniform  spherical  shell,  composed  of  any  substance 
exerting  force  which  varies  with  the  distance  according  to  the 
law  of  inverse  squares,  will  be  zero.  By  the  most  careful  ex- 
periments which  he  could  make  with  the  crude  apparatus  at 
his  disposal,  Cavendish  could  not  detect  any  evidence  of  any 
electric  force  inside  a  sphere  charged  with  electricity.  He 
therefore  concluded  that  the  elements  of  electricity  act  on  each 
other  with  forces  which  vary  inversely  with  the  square  of  the 
distance.  The  determination  of  this  law,  by  this  method,  in- 
volves the  assumption  that  the  force  exerted  between  two  quan- 
tities of  electricity  is  proportional  to  their  product.  Cavendish 
never  published  his  proof  of  this  law,  and  it  remained  unknown 
until  his  papers  were  edited  by  Maxwell.  The  general  method 
used  by  Cavendish  has  been  employed  with  the  most  sensitive 
apparatus,  and  the  law  of  electric  force  has  been  determined 
with  great  exactness. 

The  law  of  electric  force  became  known  from  the  work  of 
Coulomb  in  1785.  In  his  investigations,  Coulomb  employed  the 


274  STATIC   ELECTRICITY. 

torsion  balance  already  described  in  §159.  By  means  of  this 
instrument,  he  measured  the  repulsions  between  two  similarly 
charged  spheres  at  different  distances  from  each  other,  and 
found  that  the  force  between  them  varied  inversely  with  the 
square  of  the  distance  between  their  centres.  It  had  been 
proved  by  Newton,  that  the  force  at  any  point  outside  a  spher- 
ical shell  made  of  a  substance  which  exerts  force  according  to 
the  law  of  inverse  squares,  will  be  inversely  as  the  square  of 
the  distance  of  the  outside  point  from  the  centre  of  the  sphere. 
The  measurements  of  Coulomb  were  therefore  consistent  with 
the  hypothesis  that  each  element  of  the  electric  charge  repels 
every  other  element  with  a  force  which  varies  inversely  as  the 
square  of  the  distance.  A  similar  law  was  proved  for  the 
.Attraction  between  two  spheres  oppositely  charged. 

By  observing  the  force  between  two  electrified  spheres  at  a 
definite  distance,  and  comparing  it  with  the  force  exerted  at 
the  same  distance  between  the  spheres  when  the  charge  on  one 
of  them  had  been  halved  by  touching  the  sphere  to  another 
of  the  same  size,  Coulomb  showed  that  the  force  is  propor- 
tional to  the  magnitudes  of  the  charges. 

The  law  of  electric  force  may  therefore  be  stated  by  saying 
that  the  force  between  two  small  electric  charges  is  propor- 
tional to  the  product  of  the  charges  and  inversely  proportional 
to  the  square  of  the  distance  between  them.  Owing  to  the  fact 
that  the  force  between  two  charges  depends  on  the  medium 
which  surrounds  them,  we  assume  in  this  definition  that  the 
charges  are  in  a  vacuum.  The  force  between  the  charges  in 
air  is  only  slightly  smaller  than  that  in  vacuum. 

174.  Quantity  of  Electricity. — In  measuring  quantities  of 
electricity  we  find  ourselves  in  the  same  position  as  when  we 
measure  quantities  of  magnetism.  We  do  not  perceive  elec- 
tricity directly,  or  by  any  other  effects  than  the  forces  which 
it  exerts.  We  are  therefore  compelled  to  measure  it  by  means 
of  those  forces.  If  we  represent  two  electric  charges  by  e  and 
e',  the  distance  between  them  by  r,  and  the  factor  of  proportion 
by  fe,  the  equation  which  expresses  the  force  between  these 

charges  is  F=fc— .     We  can  measure  the  force  and  the  dis- 


STATIC   ELECTRICITY.  275 

tance,  but  until  some  convention  is  made,  we  cannot  measure 
the  charges.  It  is  conceivable  that  a  charge  might  be  arbitrar- 
ily selected  as  a  standard,  with  which  other  charges  might  be 
measured  by  the  examination  of  the  forces  which  they  exert  on 
the  standard  charge.  But  the  impossibility  of  preserving  such 
a  charge  unaltered,  and  of  using  it  in  the  study  of  other 
charges  without  error,  makes  it  necessary  for  us  to  determine 
the  unit  charge  in  some  other  way.  If  we  were  to  use  this 
method,  the  factor  k  would  have  a  definite  value,  which  might 
be  determined  by  experiment  as  the  constant  of  electric  force. 
But  so  long  as  the  unit  charge  is  left  undetermined,  the  value 
of  k  is  not  fixed.  By  assuming  an  arbitrary  value  for  it,  we 
fix  the  product  of  the  two  charges  which  act  on  each  other.  To 
make  this  value  as  simple  as  possible,  we  set  fc=l.  When  this 
choice  is  made,  the  value  of  the  product  ce'  is  determined  from 
the  known  values  of  the  force  and  the  distance.  If  the  charges 
are  equal,  the  value  of  either  charge  is  determined  from  the 
same  data.  In  particular,  we  may  define  the  unit  electric 
charge  as  that  charge  which  will  repel  an  equal  and  similar 
charge  at  unit  distance  with  unit  force.  In  the  C.  G.  S.  system 
the  unit  electric  charge  is  one  which  will  repel  an  equal  and 
similar  charge  at  the  distance  of  one  centimetre  with  the  force 
of  one  dyne. 

175.  Electric  Fields. — In  order  to  study  the  electric  forces 
exerted  by  any  system  of  electrified  bodies,  it  is  convenient  to 
suppose  that  we  can  make  use  of  a  unit  charge,  which  is  per- 
fectly insulated,  and  the  force  upon  which  we  can  always  meas- 
ure. This  charge  may  be  called  a  test  unit.  When  such  a  test 
unit  is  brought  near  an  electrified  body,  it  is  generally  acted 
on  by  an  electric  force.  The  region  in  which  such  a  force  can 
be  detected  is  called  an  electric  field. 

If  the  directions  of  the  forces  at  different  points  in  an  elec- 
tric field  are  examined  by  means  of  the  test  unit,  it  will  be 
found  that  they  vary  from  point  to  point  in  such  a  way  that 
they  may  be  represented  by  drawing  lines  of  force  through  the 
electric  field.  A  line  of  force  is  a  line  such  that  the  tangent  to 
it  at  any  point  indicates  the  direction  of  the  electric  forae  at 


276  STATIC    ELECTRICITY. 

that  point.  The  positive  direction  of  a  line  of  force  is  the 
direction  in  which  the  test  unit  will  tend  to  move.  It  will  be 
shown  later  that  the  lines  of  force  in  any  electric  field  begin  at 
positive  charges  and  end  at  negative  charges.  We  shall  also 
find  it  possible  to  connect  the  number  of  lines  of  force  with  the 
charges  at  which  they  arise  in  such  a  way  that  the  distribution 
of  the  lines  in  the  field  indicates  the  force  on  the  test  unit  at 
each  point.  This  force  at  any  point  is  called  the  strength  of 
the  electric  field  at  that  point,  or  the  electric  intensity,  or 
sometimes  the  electric  force. 

When  an  insulated  conductor  is  placed  in  an  electric  field, 
it  becomes  electrified  by  induction.  We  may  describe  the  dis- 
tribution of  the  charge  in  it,  in  a  general  way,  by  saying  that 
both  the  positive  and  the  negative  charges  move  in  the  con- 
ductor under  the  action  of  the  electric  field  as  if  they  were 
free.  The  positive  charge  is  therefore  displaced  in  the  positive 
direction  of  the  lines  of  force,  the  negative  charge,  in  the  nega- 
tive direction.  This  displacement  continues  until  the  forces 
set  up  by  the  displaced  charges,  combined  with  the  electric 
force  of  the  field,  neutralize  each  other  at  every  point  within 
the  conductor.  When  this  condition  is  reached,  the  displace- 
ment of  the  charges  ceases  and  the  system  is  in  electric  equi- 
librium. 

176.  Electric  Potential. — In  his  study  of  the  law  of  gravi- 
tation, Laplace  introduced  a  certain  function,  by  which  his 
analysis  was  very  greatly  facilitated.  A  similar  function  may 
be  used  in  the  study  of  any  forces  which  vary  according  to  the 
law  of  inverse  squares.  Such  a  function  was  applied  to  the 
study  of  electricity  by  Green  and  was  called  by  him  the  poten- 
tial function.  In  the  case  of  electricity  the  potential  function 
is  found  to  be  especially  useful,  and  to  be  applicable  even  in  an 
elementary  treatment  of  the  subject.  Indeed  our  modern  study 
of  electricity  depends  upon  its  use.  It  is  accordingly  necessary 
at  this  point  to  examine  its  most  important  properties.  We 
shall  use  the  word  potential  to  designate  not  only  this  func- 
tion, but  the  property  of  the  electric  field  characterized  by  it. 

If  a  test  unit  is  placed  at  a  point  in  an  electric  field,  and 


STATIC   BLKCTBICITY.  277 

is  moved  from  that  point  to  an  infinite  distance,  work  is  done 
upon  it  by  the  forces  of  the  field  during  its  motion.  Analysis 
proves  that  the  work  which  is  thus  done  depends  only  on  the 
position  of  the  point  at  which  the  motion  begins,  and  is  inde- 
pendent of  the  path  through  which  the  test  unit  moves.  The 
work  thus  done  is  equal  to  and  measures  the  potential  at  the 
point.  If  the  work  done  by  the  forces  of  the  field  during  the 
motion  is  positive,  the  potential  at  the  point  is  positive;  if  the 
work  done  is  negative,  the  potential  is  negative. 

If  two  points  in  the  electric  field  are  chosen,  the  work 
done  on  the  test  unit,  as  it  moves  from  one  of  them  to  the 
other,  is  the  difference  of  potential  between  those  points. 
This  difference  of  potential  is  independent  of  the  path  tra- 
versed by  the  test  unit  during  the  motion.  This  may  be  shown 
by  analysis,  and  must  otherwise  be  true  if  the  principle  of  the 
conservation  of  energy  holds  for  the  electric  field.  For  if  it 
were  not  true,  and  if  more  work  were  done  on  the  test  unit 
in  one  path  than  in  another,  an  unlimited  supply  of  work 
might  be  obtained  by  allowing  the  test  unit  to  move,  under  the 
action  of  the  field,  over  the  path  in  which  the  greater  amount 
of  work  is  done,  and  bringing  it  back,  against  the  action  of 
the  field,  over  the  other  path,  and  by  repeating  this  cyclic  pro- 
cess as  often  as  we  please.  This  result  is  of  course  incon- 
sistent with  the  principle  of  the  conservation  of  energy. 

If  the  distance  between  the  two  points,  between  which  the 
test  unit  moves,  is  very  small,  the  force  on  the  test  unit  will 
be  appreciably  constant  during  the  motion.  If  we  represent 
the  force  on  the  test  unit  in  the  direction  of  the  motion  by 
F,  the  small  distance  traversed  by  s,  and  the  potentials  at  the 
two  points  by  V  and  V,  we  obtain,  from  the  relation  between 
the  work  done  on  the  test  unit  and  the  difference  of  potential,  the 
equation  Fs=V — V.  From  this  equation  we  obtain  for  the 

V-  V. 
force  in  the  direction  of  the  motion  the  expression  F=  — — 

The  quantity  on  the  right  is  evidently  the  change  of  potential 
in  the  direction  of  the  motion  per  unit  of  length,  or  is  the  rate 
of  change  of  potential  in  the  direction  of  motion.  In  the 
electric  field  the  force  is  positive,  or  is  in  the  direction  of  the 


"2~S  STATIC    KLKOTRICITT. 

motion,  when  the  unit  moves  from  the  point  of  higher  poten- 
tial to  the  point  of  lower  potential.  When  there  is  no  differ- 
ence of  potential  between  the  neighboring  points,  there  is  no 
force  acting  in  the  line  joining  them.  In  an  electric  field  a 
system  of  surfaces  can  be  drawn,  which  are  everywhere  at 
right  angles  to  the  lines  of  force.  There  is  no  component  of 
force  lying  in  any  of  these  surfaces,  and  consequently  in  no 
one  of  them  is  there  any  difference  of  potential.  Any  such 
surface,  in  which  the  potential  has  everywhere  the  same  value, 
is  called  an  equipotential  surface.  From  this  relation  it  is 
evident  that  any  field  of  force,  which  can  be  mapped  out  by 
lines  of  force,  can  also  be  mapped  out  by  equipotential  sur- 
faces. 

The  peculiar  applicability  of  the  potential  to  the  study  of 
static  electricity  is  due  to  the  fact  that  electricity  moves 
freely  through  a  conductor  so  long  as  there  is  any  electric 
force  acting  on  it,  and  attains  equilibrium  only  when  its  dis- 
tribution is  such  that  no  electric  forces  are  acting  within  the 
conductor.  In  this  condition,  as  is  evident  from  the  relation 
between  force  and  change  of  potential,  the  potential  of  the 
conductor  must  be  everywhere  the  same,  and  the  surface  of 
the  conductor  must  be  an  equipotential  surface.  The  potential 
at  any  point  on  this  surface  is  called  the  potential  of  the 
conductor. 

In  order  to  measure  the  potential  of  a  conductor  accord- 
ing to  the  definition  of  it  which  has  been  given,  it  would  be 
necessary  to  carry  a  test  unit  from  that  conductor  to  infinity. 
By  this  operation  we  measure  the  difference  of  potential  be- 
tween the  conductor  and  an  infinitely  distant  point.  Of 
course  this  operation  is  impossible,  and  it  is  therefore  neces- 
sary, for  practical  purposes,  to  adopt  the  potential  of  some 
other  point  as  the  standard  potential,  with  which  other 
potentials  shall  be  compared.  The  standard  potential  chosen 
is  the  potential  of  the  earth,  which,  it  may  be  shown,  re- 
mains appreciably  constant  at  all  times,  whatever  be  the  elec- 
tric operations  which  take  place  during  our  experiments. 
This  standard  potential  we  take  as  the  zero  of  potential. 


STATIC   ELECTRICITY. 


279 


With  this  convention  the  potential  of  a  conductor  is  the  work 
which  is  done  on  a  test  unit  as  it  moves  from  the  conductor  to 
the  earth.  The  potential  of  a  positively  charged  body  is  posi- 
tive; that  is,  the  work  which  is  done  on  the  test  unit,  aa  it 
moves  from  that  body  to  the  earth,  is  positive  work  done  by 
the  electric  forces.  The  potential  of  a  negatively  charged  body 
is  negative;  that  is  the  work  done  by  the  electric  forces  on  the 
test  unit,  as  it  is  moved  from  the  body  to  the  earth,  is  nega- 
tive. These  statements  do  not  apply  without  exception  to 
bodies  on  which  there  are  induced  charges. 

The  tendency  of  a  positive  charge  is  to  pass  from  a  place 
of  higher  potential  to  a  place  of  lower  potential.  The  ten- 
dency of  a  negative  charge  is  to  pass  from  a  place  of  lower 
potential  to  a  place  of  higher  potential.  When  a  caarged 
body  is  brought  near  enough  to  a  conductor  whose  potential 
differs  from  its  own,  its  charge  will  pass  in  the  form  of  a 
spark  to  the  other  conductor,  until  the  potentials  of  the  two 
conductors  become  the  same. 

We  may  illustrate  the  application  of  the  idea  of  potential 
by  describing,  in  terms  of  it,  the  use  of-  the  electrophorus. 
When  the  plate  of  the  electrophorus  is  charged  by  friction, 
its  potential,  and  that  of  the  region  around  it,  becomes  nega- 
tive. The  numerical  value  of  this  negative  potential  is  high- 
est at  the  plate,  and  diminishes  from  the  plate  to  zero.  When 
the  carrier  is  placed  on  the  plate,  it  is  brought  into  this 
region  of  negative  potential,  and  if  it  were  not  a  conductor, 
the  potential  on  its  lower  surface  would  be  negatively  higher 
than  the  potential  on  its  upper  surface.  There  would  thus 
exist  in  it  a  difference  of  potential,  and  the  lines  of  force  cor- 
responding to  this  difference  would  be  directed  from  its  upper 
surface  to  its  lower  surface.  Such  a  force,  however,  cannot 
exist  in  a  conductor,  and  a  positive  charge  is  developed  on  the 
lower  surface,  and  a  corresponding  negative  charge  on  the 
upper  surface,  until  the  forces  to  which  they  give  rise 
neutralize  the  force  which  has  been  described,  or  until  the 
potential  within  the  carrier  is  everywhere  the  same.  When 
the  carrier  is  touched  and  so  joined  to  earth,  its  potential  is 


280  STATIC   ELECTRICITY. 

raised  to  zero,  the  potential  of  the  earth.  This  involves  the 
loss  of  the  negative  charge  of  the  carrier.  The  potential  of 
the  carrier  is  then  due  to  the  superposition  of  the  opposite 
potentials  due  to  the  charge  on  the  plate  and  the  positive 
charge  on  the  carrier.  When  the  carrier  is  lifted,  work  is  done 
on  it  against  the  attraction  between  these  charges,  and  the 
work  thus  done  raises  the  electric  energy  of  the  carrier,  and 
gives  it  positive  potential. 

The  potential  of  a  conductor  is  measured  by  the  work  done 
on  unit  charge,  as  it  is  carried  from  the  conductor  to  the  earth. 
The  potential  of  a  conductor  is  therefore  equal  to  unity,  or  the 
conductor  is  at  unit  potential,  when  the  work  done  on  the 
unit  charge  is  the  unit  quantity  of  work.  In  the  C.  G.  S. 
system  the  potential  of  a  conductor  is  unity  when  the  work 
done  on  the  unit  charge  of  that  system,  as  it  passes  from  the 
conductor  to  the  earth,  is  equal  to  one  erg. 

177.  Equality  of  the  Two  Kinds  of  Electricity. — It  was 
assumed  by  Dufay  and  Franklin,  and  accepted  by  all  other 
students  of  the  subject,  that  the  two  electric  conditions,  called 
vitreous  and  resinous,  or  positive  and  negative,  existed  in 
every  natural  body  in  equal  quantities.  Our  attention  has 
already  been  called  to  this  assumption  in  our  study  of  in- 
duction, in  Avhich  we  saw  that  the  two  charges  which  are 
separated  by  induction  will  exactly  neutralize  each  other  when 
the  conductor  is  removed  from  the  electric  field. 

It  may  also  be  shown  by  experiment  that  when  electricity 
is  produced  by  friction,  it  appears  in  equal  quantities  of  the 
two  kinds.  For  example,  if  a  rod  of  gla«s  is  fitted  with  a  cap 
of  silk  or  flannel,  held  in  an  insulating  handle,  and  is  turned 
in  the  cap  until  its  end  is  electrified,  it  may  be  shown  that  the 
cap  is  also  electrified  oppositely;  and  the  equality  of  the  two 
charges  is  shown  by  the  fact  that  when  the  cap  is  on  the  glass 
rod,  no  electrical  effects  can  be  detected.  This  experiment  suc- 
ceeds best  if  the  glass  and  silk  are  rubbed  together  inside  a 
closed  metallic  vessel  insulated  from  the  earth.  As  we  shall 
see  in  the  next  paragraph,  the  exterior  of  this  vessel  ought  to 
appear  electrified  unless  the  two  charges  developed  are  ex- 


STATIC   ELECTRICITY.  281 

actly  equal.  That  two  opposite  charges  are  developed,  is 
shown  by  removing  either  the  glass  or  the  silk  from  the  vessel, 
the  exterior  of  which  then  becomes  either  negatively  or  posi- 
tively electrified.  But  when  the  glass  and  silk  are  in  the 
vessel  together,  no  external  electrification  can  be  detected. 
Indeed,  any  electric  machine,  or  the  most  intricate  methods 
for  developing  electricity,  may  be  worked  inside  the  vessel 
without  electrifying  its  exterior  at  all. 

The  final  demonstration  of  the  equality  between  the  posi- 
tive and  negative  charges,  and  of  the  invariable  relation  be- 
tween them,  was  given  by  Faraday.  The  experiment  by  which 
Faraday  did  this  is  commonly  called  the  ice  pail  experiment, 
because  the  vessels  which  he  used  in  it  were  ice  pails.  In  its 
simplest  form,  the  apparatus  consists  of  a  metallic  vessel,  set 
on  an  insulating  stand,  and  of  an  insulated  conductor,  which 
can  be  introduced  into  the  interior  of  the  vessel.  The  cover 
of  the  vessel  is  put  on  after  this  conductor  is  introduced. 
Some  sensitive  apparatus  is  provided  by  which  the  electrifica- 
tion of  the  exterior  of  the  vessel  can  be  examined.  The  vessel 
is  first  discharged  by  joining  it  to  the  earth  for  a  moment, 
and  the  conductor,  charged  positively,  is  introduced  within  it. 
The  vessel  is  then  found  to  be  charged  positively.'  If  it  is 
then  joined  to  earth  for  a  moment,  its  positive  charge  dis- 
appears. If  the  charged  conductor  is  now  removed  from  the 
interior,  the  vessel  exhibits  a  negative  charge.  Judging  from 
what  we  have  already  learned  about  the  distribution  of  in- 
duced charges,  we  may  conclude  that  the  effect  of  introducing 
the  charged  conductor  within  the  vessel  is  to  develop  a  nega- 
tive charge  on  its  inner  surface  and  a  positive  charge  on  its 
outer  surface.  These  charges  are  equal  in  magnitude,  as  is 
proved  by  their  neutralizing  each  other,  if  the  charged  con- 
ductor is  removed  before  the  positive  charge  of  the  vessel 
has  been  removed  by  joining  it  to  earth.  Now  to  determine 
the  relative  magnitudes  of  the  original  charge  on  the  con- 
ductor and  the  opposite  charge  which  it  produces  by  induction, 
we  introduce  the  charged  conductor  as  before,  and  remove  the 
induced  positive  charge.  We  then  touch  the  charged  conductor 


282  STATIC    ELKCTRLCITY. 

to  the  interior  of  the  vessel,  or  join  it  to  the  interior  by  a 
conducting  wire.  No  signs  of  electrification  appear  on  the 
exterior,  and  when  the  conductor  is  now  removed,  no  charges 
can  be  detected  either  on  it  or  on  the  vessel.  Faraday  con- 
cluded from  this  result  that  the  charge  on  a  body  induces  a 
charge  of  the  opposite  sort,  and  of  equal  magnitude  to  itself, 
on  the  conductor  or  conductors  which  completely  surround  it. 

Before  the  significance  of  this  experiment  was  appreciated, 
it  was  usual  to  speak  of  charged  bodies  which  are  at  some  dis- 
tance from  other  conductors  as  freely  electrified  bodies,  and 
to  ignore  the  opposite  charges  which  they  produce  in  neigh- 
boring bodies  by  induction.  It  was  only  when  the  charged 
body  is  very  near  other  conductors,  as  in  the  case  of  the 
Leyden  jar,  that  the  presence  of  the  induced  charge  was 
thought  of  as  having  any  special  significance.  Faraday's  ex- 
periment showed,  however,  that  there  is  really  no  such  thing 
as  a  freely  electrified  body,  but  that  the  electrification  of  one 
body  involves  the  electrification  by  induction  of  the  conductors 
which  surround  it.  This  general  truth  suggests  the  hypothesis 
that  the  two  equal  charges  which  confront  each  other  across 
the  non-conducting  medium  which  separates  them,  are  the  two 
ends  of  a  condition  which  exists  in  the  medium,  and  that  the 
medium  is  the  true  seat  of  the  electric  action.  We  shall  find 
additional  support  for  this  hypothesis  when  we  study  the  way 
in  which  different  media  affect  the  charge  which  a  body  can 
receive. 

178.  Capacity  of  Conductors. — We  define  the  capacity  of 
a  conductor  as  the  quantity  of  electricity  which  is  required  to 
raise  the  potential  of  the  conductor  from  zero  to  unity,  when 
th  potential  of  all  surrounding  conductors  is  maintained  at 
zero.  The  capacity  of  a  conductor  depends,  at  least  in  part, 
on  its  shape  and  size,  and  on  the  distance  between  it  and  sur- 
rounding conductors.  If  we  examine  the  explanation  which 
has  been  given  in  §170  of  the  charging  of  a  Leyden  jar,  it  will 
be  seen  that  the  mutual  inductive  action  between  the  two 
coatings  of  the  jar  will  be  greater  when  they  are  nearer  to- 
gether, and  that  consequently  a  larger  charge  will  be  received 


STATIC    KLKOTR1CITY.  288 

by  the  inner  coating  from  the  same  source,  when  the  coatings 
are  near  together,  than  when  they  are  farther  apart.  This 
general  conclusion  may  be  formally  demonstrated  in  the  case 
of  the  spherical  condenser.  The  spherical  condenser  is  a  con- 
ducting sphere  insulated  from  a  concentric  spherical  conduct- 
ing shell.  The  outer  shell  is  joined  to  earth,  and,  through  a 
small  hole  in  it,  contact  is  made  by  means  of  a  wire  between 
the  interior  sphere  and  the  source  of  charge.  The  potential  of 
the  interior  sphere  is  everywhere  the  same,  and  we  can  cal- 
culate it  by  calculating  the  potential  at  its  centre.  It  may 
be  shown  that  the  potential  at  a  point  at  the  distance  r  from 

a     chanje   of  magnitude   e    is   equal    to    -•       Now  all  parts    of 

the  charge  on  the  interior  sphere  are  at  the  same  distance 
from  the  centre  of  the  sphere,  and  the  potential  at  that  cen- 
tre due  to  the  charge  c  on  the  sphere  is  equal  to  -  ,  in  which 

r  represents  the  radius  of  the  sphere.  The  potential  at  the 
same  point,  due  to  the  equal  and  opposite  charge  on  the  inner 

surface  of  the  exterior  shell,  is  — e .,  in  which  »•'  represents 

the  radius  of  the  shell.  The  actual  potential  at  the  centre  of 
the  sphere,  and  therefore  of  the  sphere  itself,  is  the  sum  of 

/  Tf T  \ 

these  potentials,  so  that  we  may  write  P=e|  — —  )  The  ratio 

of  the  charge  to  the  potential,  or -^—,   is  the  capacity  of  the  in- 

rf — r 

terior  sphere,  as  we  have  already  denned  it.  As  the  formula 
shows,  it  is  equal  to  the  product  of  the  two  radii  divided  by 
their  difference,  and  therefore  increases  very  rapidly  as  the 
difference  of  the  radii  is  diminished. 

A  system  of  conductors  like  the  spheres  just  described,  or 
like  the  Leyden  jar,  of  which  the  capacity  is  very  great,  is 
called  a  condenser.  The  conductor  which  is  charged  in  a 
condenser  differs  from  other  conductors  merely  in  being  so 
situated,  with  respect  to  the  conductors  which  surround  it, 
that  a  relatively  large  charge  is  required  to  raise  it  to  unit 
potential. 

It  is  obvious  that  the  conclusion  which  has  been  proved  to 


284  STATIC   ELECTRICITY. 

hold  for  the  spherical  condenser,  that  its  capacity  depends 
upon  the  distance  between  the  charged  body  and  the  con- 
ductors which  surround  it,  will  hold  for  conductors  of  any 
shape. 

179.  The  Dielectric. — An  isolated  experiment  tried  by 
Franklin  led  him  to  believe  that  the  non-conducting  medium 
between  the  two  coatings  of  a  Leyden  jar  plays  an  important, 
and  perhaps  the  principal  part,  in  the  charging  of  the  jar. 
We  now  try  this  experiment  with  what  is  known  as  a  dissected 
jar,  that  is,  a  glass  vessel  or  cup  which  is  set  inside  a  metallic 
cup,  to  serve  as  an  outer  coating,  and  receives  another  metallic 
cup  furnished  with  a  conducting  rod,  to  serve  as  the  inner 
coating.  Such  a  jar  may  be  charged  and  discharged  in  the 
ordinary  way.  If  the  inner  coating  is  removed  while  the  jar 
is  charged,  it  will  be  found  to  have  only  a  very  small  charge  on 
it.  It  appears,  therefore,  that  the  charge  of  the  jar  is  not 
carried  about  on  the  inner  coating.  If  this  coating  is  re- 
placed, the  full  charge  may  be  obtained  from  the  jar.  The  ex- 
periment is  more  striking  if  we  remove  both  the  coatings  and 
replace  them  by  others  of  similar  shape.  The  jar  is  still  found 
to  be  charged,  and  the  conclusion  is  irresistible  that  the  charge 
has  resided  somewhere  in  or  on  the  glass.  It  is  natural  to 
make  the  hypothesis  that  the  process  of  electrification  con- 
sists in  setting  up  in  the  glass  some  peculiar  condition,  which 
terminates  at  the  two  conducting  coatings. 

When  a  non-conducting  medium  is  used  to  separate  a 
charged  conductor  from  the  other  conductors  which  surround 
it,  and  when  we  are  studying  the  conditions  which  exist  in 
that  medium,  or  the  influence  of  the  medium  upon  the  charge, 
we  call  the  medium  the  dielectric.  The  first  study  of  the 
properties  of  dielectrics  was  made  by  Cavendish  (1771-1781), 
but  his  work  was  not  published  at  that  time,  and  we  owe  our 
knowledge  of  those  properties  to  Faraday.  Faraday's  in- 
vestigation was  carried  out  by  the  aid  of  two  precisely  simi- 
lar spherical  condensers.  In  one  of  these  the  space  between 
the  two  surfaces  was  always  filled  with  air,  in  the  other  it 
was  first  filled  with  air,  and  afterwards  with  the  different 


STATIC   ELECTRICITY.  285 

dielectrics  which  he  studied.  The  experiment  consisted  in 
charging  the  inner  spheres  of  both  these  condensers  to  the 
same  potential,  by  joining  them  both  at  the  same  time  to  the 
same  source,  and  in  discharging  them  in  turn  through  an 
instrument  by  means  of  which  the  quantities  discharged  could 
be  compared.  As  was  to  be  expected  from  the  similarity  of  the 
condensers,  the  quantity  discharged  was  the  same  from  each 
when  both  contained  air  as  the  dielectric.  When  sulphur  or 
parafline  was  used  in  one  of  the  condensers  as  the  dielectric, 
the  quantity  obtained  from  it  was  many  times  greater  than 
the  quantity  obtained  from  the  other.  This  experiment  proved 
that  the  capacity  of  a  condenser  depends  on  the  nature  of  the 
dielectric  which  separates  its  conducting  parts.  Faraday  used 
the  term  specific  inductive  capacity  to  represent  the  effect  of 
a  particular  dielectric,  as  determined  by  the  ratio  of  the 
capacity  of  a  condenser  in  which  it  is  used  to  the  capacity  of  a 
similarly  shaped  condenser  in  which  air  is  used.  This  ratio, 
which  is  a  characteristic  constant  for  each  particular  di- 
electric, is  now  generally  called  the  dielectric  constant. 

This  direct  and  very  complete  evidence  of  the  participation 
of  the  dielectric  in  the  process  of  charging  a  body  confirmed 
the  hypothesis  which  has  already  been  referred  to,  that  the 
production  of  the  charge  involves  setting  up  in  the  dielectric  a 
peculiar  condition,  which  terminates  at  the  charged  con- 
ductors. This  hypothesis  has  been  worked  out  analytically 
by  Maxwell,  and  has  been  shown  to  give  a  complete  account  of 
the  relations  of  charged  bodies. 

On  this  view  of  the  process  of  electrification,  the  lines  of 
force  which  may  be  drawn  through  the  dielectric  are  naturally 
taken  to  indicate  lines  along  which  the  action  in  the  dielec- 
tric takes  place.  We  may  map  out  the  electric  field,  and  in- 
dicate its  intensity  at  various  points,  by  drawing  one  line 
of  force,  or  any  assumed  number  of  lines  of  force,  from  each 
unit  of  positive  charge,  through  the  dielectric,  to  the  cor- 
responding unit  of  negative  charge.  The  measurements  of  the 
electric  force  from  which  these  lines  are  drawn  must  be  made 
in  a  narrow  crevasse  cut  in  the  dielectric,  with  its  faces  per- 


286  STATIC   ELECTRICITY. 

pendicular  to  the  lines  of  force.  The  force  thus  determined 
is  called  the  electrostatic  induction,  and  the  lines  are  called 
lines  of  electrostatic  induction.  It  may  be  shown  that  where 
these  lines  of  induction  are  most  closely  crowded  together,  the 
electric  intensity  is  greatest,  and  that  the  number  of  lines  of 
induction,  which  at  any  point  pass  perpendicularly  through  a 
unit  of  area,  will  be  proportional  to  the  intensity  at  that 
point. 

180.  Maxwell's  Descriptive  Theory  of  Electrification. — In 
connection  with  the  theory  of  medium  action,  it  is  interesting 
to  examine  the  particular  form  given  it  by  Maxwell,  rather 
as  a  description  than  as  expressing  any   final  theory  of  the 
real  condition  in  a  dielectric.     Maxwell   supposed  the   ether, 
and  therefore  all  bodies  in  the  ether,  to  be  filled  with  elec- 
tricity.    We  may  best  think  of  this  electricity  as  existing  in 
separate    portions.      Maxwell    supposed    that    this    electricity 
moves    freely,    or    with    a    resistance    due    only    to    friction, 
through  conductors,  but  that  its  displacement  in  a  dielectric 
is  resisted  by  a  force  which  increases  with  the  extent  of  the 
displacement,  and  which  he  likened  to  elasticity.     The  process 
of  charging  a  body  then  consists,  according  to  this  description, 
in  a  displacement  of  the  electricity  along  the  lines  of  force 
until  it  is  checked  by  the  electric  elasticity.     Considering  the 
displacement  with  respect  to  the  dielectric,  it  is  inward  when 
the  dielectric   is  bounded  by   a   positively   charged   conductor 
and  outward  when  it  is  bounded  by  a  negatively  charged  con- 
ductor.    When   connection   is  made  by  a   conductor  between 
two   oppositely    charged   bodies,   a    flow   of   electricity    passes 
through  it  and  the  strain  in  the  dielectric  is  relieved.     The 
work  which  is  done  in  charging  a  conductor  is,  in  this  descrip- 
tion, stored  up  in  the  dielectric  as  work  done  in  effecting  the 
electric  displacement.     To  account  for  the  various  dielectric 
constants  of  different  media,  we  suppose  that  the  same  force 
will  produce  different  displacements  in  the  different  media. 

181.  Electric   Machines. — The  first  electric   machine,   con- 
structed  by   Von   Guericke,   has   already   been    described.      It 
was    very    soon    improved    upon    by    using    a    glass    plate    or 


STATIC   ELECTRICITY.  287 

cylinder,  instead  of  the  sulphur  ball,  and  by  setting  against  it 
a  pad  of  flannel  or  leather,  by  the  friction  of  which  the  glass 
is  electrified.  An  insulated  conductor,  called  the  prime  con- 
ductor, furnished  with  a  comb  or  row  of  points,  pointing  to- 
ward the  glass,  is  used  to  collect  the  charge  developed  as  the 
machine  is  turned.  Machines  of  this  sort  are  called  frictional 
machines.  They  are  now  very  little  used. 

The  germ  of  all  "modern  electric  machines  is  found  in  the 
electrophorus  of  Volta.  It  is  plain  that  if  the  operation  of 
charging  and  discharging  the  carrier,  which  we  have  described 
as  carried  on  by  the  hand,  is  executed  rapidly  by  some  mechan- 
ism, a  rapid  succession  of  charges  can  be  obtained.  It  is  not 
necessary  to  go  into  the  details  of  these  induction  machines. 

182.  Electroscopes  and  Electrometers. — Any  apparatus 
which  will  indicate  the  presence  of  an  electric  charge,  and 
which  will  enable  us  to  determine  its  character,  is  called  an 
electroscope.  Any  light  body,  such  as  a  straw  suspended  by 
a  silk  thread,  may  serve  for  that  purpose.  A  very  common 
form  of  electroscope  is  that  known  as  the  gold  leaf  electro- 
scope. It  consists  of  two  narrow  strips  of  gold  foil,  which 
hang  down  from  the  end  of  a  conducting  rod,  supported  in  the 
neck  of  a  glass  bottle  or  flask.  When  this  system  is  charged, 
the  gold  leaves,  being  charged  similarly,  mutually  repel  each 
other,  and  diverge  from  each  other.  They  are  most  commonly 
charged  positively  by  induction,  using  a  negative  primary 
charge.  While  the  gold  leaves  are  diverging  because  of  their 
positive  charges,  if  the  upper  end  of  the  rod  to  which  they  are 
attached  is  brought  near  a  positively  charged  body,  an  addi- 
tional positive  charge  will  be  given  to  them  by  induction, 
and  they  will  diverge  more  widely.  If,  on  the  other  hand,  the 
body  to  which  the  rod  is  presented  is  charged  negatively,  the 
gold  leaves  will  receive  a  negative  charge  by  induction,  which 
will  neutralize  some  of  their  positive  charge,  and  they  will 
diverge  less  widely.  This  apparatus,  without  any  charge,  is  a 
very  sensitive  indicator  of  the  presence  of  a  charge  on  any 
body  to  which  it  is  brought  near. 

An  electrometer  is  an  instrument  by  which  the  difference 
of  potential  between  two  bodies  can  be  measured,  or  compared 


288  STATIC    ELECTRICITY. 

with  some  other  difference  of  potential.  The  attracted-disk 
electrometer,  first  used  by  Snow  Harris,  and  developed  by  Lord 
Kelvin,  furnishes  a  measure  in  absolute  units  of  difference  of 
potential.  It  consists  essentially  of  a  large  horizontal  disk, 
above  which  a  smaller  disk  is  set  at  a  known  distance.  Thia 
smaller  disk  is  supported  by  a  balance  or  a  spring,  by  means 
of  which  the  force  upon  it  may  be  measured.  By  measuring 
the  force  exerted  by  the  charges  on  the  disks,  when  they  are 
brought  to  different  potentials,  the  difference  of  potential  be- 
tween them  may  be  measured  in  absolute  units. 


THE   BLKCTRIC  CURRENT. 


THE  ELECTRIC  CURRENT. 

183.  Galvani's  Discovery. — In  1791  the  Italian  physiologist 
Galvani  happened  to  notice  that  the  legs  of  a  recently  killed 
frog,  lying  near  an  electric  machine,  were  thrown  into  con- 
vulsions whenever  a  spark  passed  from  the  machine.     He  in- 
vestigated this  phenomenon  on  the  hypothesis  that  the  nerves 
and  muscles  act  like  the  coatings  of  a  charged  Leyden  jar,  and 
in  following  up  this  hypothesis  he  found  that  the  convulsive 
movements    occurred    whenever    the    lumbar    nerves    and    the 
muscles  of  the  leg  were  joined  by  a  metallic  connection.    The 
action,  which  was  comparatively  slight  when  this  connection 
was  made  by  one  metal,  was  made  much  greater  by  touching 
the  nerve  with  a  strip  of  one  kind  of  metal,  the  muscles  with 
a  strip  of  another  kind,  and  then  bringing  the  two  strips  to- 
gether.    Galvani  interpreted  this  result  consistently  with  the 
hypothesis   already   stated,   but  other  observers  were   led  to 
consider  the  frog's  legs  simply  as  a  very  sensitive  electroscope, 
and  to  ascribe  the  action  observed  to  the  contact  of  the  metals. 

184.  Volta's  Series.— In  the  years   1798-1802   the  Italian 
physicist  Volta  succeeded  in  demonstrating  the  production  of 
electrification  by  the  contact  of  metals,  and  in  applying  his 
discovery  to   the  construction   of   an   apparatus   for   the   pro- 
duction of  the  electric  current.     By  the  use  of  the  condensing 
electroscope,  which  he  had  invented  a  few  years  before,  Volta 
was  able  to  prove  that  when  pieces  of  two  different  metals,  like 
copper  and  zinc,  are  brought  in  contact,  the  electric  condition 
of  the  one  becomes  different  from  that  of  the  other.     As  we 
should  now  describe  it,  the  potentials  of  the  two  metals  be- 
come different  from  each  other  when  the  metals  are  brought  in 
contact.     This  difference  of  potential  is  very  slight,  altogether 
too  slight  to  be  detected  by  an  observation  with  any  but  a 
sensitive  electroscope. 

Volta  demonstrated  that  the  difference  of  potential  occur- 
ring on  contact  is  a  definite  characteristic  difference  for  each 


290  THB    KLECTR1C   CUBRENT. 

pair  of  metals  employed.  He  proved  also  that  the  electric 
effect  produced  by  arranging  a  number  of  different  metals  in 
contact  with  one  another  in  succession  is  to  charge  the  two 
metals  at  the  end  of  the  row,  just  as  they  would  be  charged  if 
they  were  immediately  in  contact.  By  measuring  the  different 
potential  differences  existing  between  the  different  pairs  of 
metals,  Volta  found  that  the  algebraic  sum  of  the  potential 
differences  arising  at  the  successive  contacts  is  equal  to  the 
difference  of  potential  arising  from  the  contact  of  the  two 
terminal  metals  of  the  arrangement.  If  the  two  terminal 
metals  are  brought  in  contact,  so  that  a  closed  series  of  dif- 
ferent metals  is  formed,  the  algebraic  sum  of  the  differences 
of  potential  arising  from  the  contacts  will  equal  zero.  That 
this  is  the  case  appears  from  the  fact  that  there  is  no  evidence 
of  any  continuous  movement  of  electricity,  arising  from  an 
unbalanced  potential  difference,  in  such  a  metallic  circuit. 
Taking  some  one  metal  as  a  standard,  and  determining  the 
potential  differences  arising  by  its  contact  with  other  metals, 
we  obtain  data  from  which  the  difference  of  potential  arising 
from  the  contact  of  any  two  of  the  metals  may  be  determined. 
Such  a  set  of  data  is  called  a  Volta's  Series,  or  the  Electro- 
motive Series.  Volta  used  the  term,  conductors  of  the  first 
class,  to  designate  those  substances  which  give  rise  to  such 
potential  differences  that  their  algebraic  sum  equals  zero, 
when  the  substances  are  arranged  in  a  closed  circuit. 

Volta  could  not  discover  any  difference  of  potential  aris- 
ing from  the  contact  of  a  metal  with  a  liquid.  He  therefore 
called  the  liquids  conductors  of  the  second  class.  We  now 
know  that  slight  potential  differences  do  arise  from  such  a 
contact,  but  that  they  are  neither  of  such  a  magnitude  nor  of 
such  a  sign  as  to  reduce  the  sum  of  the  potential  differences 
in  a  circuit  to  zero,  when  a  conductor  of  the  second  class  forms 
a  part  of  the  circuit.  In  such  a  circuit,  therefore,  in  which, 
as  its  essential  characteristic,  two  different  conductors  of 
the  first  class  are  in  contact  with  a  conductor  of  the  second 
class,  there  exists  an  unbalanced  difference  of  potential.  We 
shall  see  later  that  conductors  of  the  second  class  are  always 


THE    KLECTRIC   CURRENT.  291 

substances  which  act  chemically  upon  the  conductors  of  the 
first  class,  and  that  a  liquid  like  mercury,  by  which  no  such 
chemical  action  is  exerted,  is  in  the  first  class. 

185.  The  Voltaic  Battery. — Following  out  the  indications 
.of  his  theory,  Volta  undertook  to  increase  the  potential  dif- 
ference developed  by  contact,  by  bringing  together  a  number 
of  successive  pairs  of  the  same  metals.  To  do  this  he  placed 
a  disk  of  paper  or  felt  moistened  with  water,  in  which,  to  ren- 
der it  a  better  conductor,  salt  or  acid  was  dissolved,  between  a 
disk  of  copper  and  a  disk  of  zinc.  Such  a  set  of  disks  we  may 
call  an  element.  Having  constructed  a  number  of  such  ele- 
ments, he  placed  one  above  the  other,  the  zinc  of  the  first 
element  being  in  contact  with  the  copper  of  the  second  element, 
so  as  to  form  a  column,  or,  as  it  was  called,  a  pile.  This  pile 
was  found  to  exhibit  all  the  ordinary  electric  phenomena.  It 
behaved,  as  Volta  said,  like  a  battery  of  Leyden  jars,  with 
this  difference,  that  the  jars,  when  discharged,  need  to  be 
charged  from  an  outside  source  before  another  discharge  can  be 
obtained  from  them,  whereas  the  pile  charges  itself,  so  that 
the  electric  effects  can  be  obtained  from  it  continuously. 

The  difference  of  potential  between  the  copper  wires  joined 
to  the  two  ends  of  the  pile  is  equal  to  the  difference  of  poten- 
tial due  to  the  contact  of  copper  and  zinc,  multiplied  by  the 
number  of  elements  in  the  pile.  According  to  Volta's  inter- 
pretation of  the  facts,  a  difference  of  potential  arises  between 
the  first  and  second  elements  from  the  contact  of  the  two 
metals,  and  the  potential  of  the  two  metals  in  the  same  ele- 
ment is  the  same,  by  reason  of  the  presence  between  them  of 
a  conductor  of  the  second  class.  The  third  element,  in  con- 
tact with  the  second,  introduces  another  difference  of  poten- 
tial, so  that  the  difference  of  potential  between  it  and  the 
lower  metal  of  the  first  element  is  double  that  due  to  one 
element.  Thus  each  successive  element,  and  finally  the  copper 
wire  joined  to  the  zinc  of  the  last  element,  introduces  an  addi- 
tional potential  difference. 

Volta  soon  recognized  that  a  pile  of  this  sort  is  not  so  well 
adapted  for  continuous  use  as  another  arrangement  of  the 


292  THE    ELECTRIC   CURRENT. 

conductors.  He  therefore  constructed  the  arrangement  which 
he  called  the  crown  of  cups,  and  which  we  now  call  the  voltaic 
battery.  Each  element  of  this  battery  is  called  a  voltaic  cell. 
A  typical  voltaic  cell  consists  of  a  glass  vessel  partly  filled 
with  acidulated  water,  in  which  are  immersed  a  plate  of 
copper  and  a  plate  of  zinc.  Wires  are  attached  to  the  upper 
ends  of  these  plates,  by  which  the  difference  of  potential  ex- 
isting between  them  can  be  transferred  to  any  desired  point. 
The  voltaic  battery  consists  of  a  number  of  such  cells,  of  which 
the  zinc  of  one  is  joined  by  a  wire  to  the  copper  of  the  next. 

In  the  typical  case  of  copper  and  zinc  and  of  the  cell 
formed  with  them,  it  may  be  interesting  to  notice  that  when 
zinc  and  copper  are  brought  in  contact,  the  potential  of  the 
zinc  becomes  positive  to  that  of  the  copper.  The  zinc  is 
therefore  said  to  be  electro-positive  to  copper.  When  the  zinc 
and  copper  of  the  cell  are  furnished  with  wires  of  the  same 
sort,  the  potential  of  the  wire  joined  to  the  copper  is  positive 
as  compared  with  the  potential  of  the  wire  joined  to  the 
zinc.  As  we  commonly  think  of  the  electric  current  as  flowing 
from  the  higher  to  lower  potential  in  that  part  of  the  circuit 
which  lies  outside  the  cell,  we  call  the  copper  plate  the  posi- 
tive pole  of  the  cell,  and  the  zinc  plate  the  negative  pole. 

Volta  considered  the  voltaic  battery  as  a  source  of  a  con- 
tinuous electric  discharge.  He  found  that  his  pile  would  give 
the  shock  which  is  felt  in  the  body  when  a  Leyden  jar  is  dis- 
charged through  it.  He  also  found  that  when  the  terminal 
poles  of  the  pile,  or  of  the  battery,  were  joined  by  a  thin  wire, 
the  wire  became  heated.  A  similar  heating  effect  had  been 
observed  when  a  battery  of  Leyden  jars  was  discharged 
through  a  wire.  By  the  use  of  modified  forms  of  the  battery, 
a  very  great  amount  of  heat  was  developed  in  the  circuit,  and 
it  was  shown  that  the  heat  developed  depends  in  some  way 
upon  the  nature  of  the  materials  composing  the  circuit.  The 
quantitative  law  of  the  development  of  heat  was  discovered  by 
Joule  in  1842. 

186.  Chemical  Action  of  the  Current. — When  Volta'a  ac- 
count of  the  battery  reached  England  in  1800,  two  members  of  . 


THE   ELECTRIC   CURRENT.  293 

the  Royal  Society,  Nicholson  and  Carlisle,  constructed  a  bat- 
tery and  made  experiments  with  it.  In  one  of  these  they  in- 
troduced the  ends  of  the  terminal  wires  into  a  drop  of  water, 
and  noticed  that  bubbles  of  gas  arose  from  them.  They  pro- 
ceeded to  investigate  this  phenomenon  with  an  apparatus  con- 
structed as  follows:  Two  test  tubes  filled  with  acidulated 
water  were  placed  with  the  open  ends  down  in  acidulated 
water  contained  in  a  vessel,  the  water  being  held  up  in  the 
tubes  by  atmospheric  pressure.  In  the  open  ends  of  the  tubes 
were  placed  small  platinum  plates,  to  which  were  joined  the 
terminal  wires  of  the  battery,  these  wires  being  covered  with 
insulating  material,  so  that  the  current  could  not  enter  the 
water  except  from  the  platinum  plates.  When  the  circuit,  was 
completed,  gases  were  evolved  at  both  the  platinum  terminals, 
and  collected  in  the  upper  ends  of  the  tubes.  These  gases, 
when  examined,  were  found  to  be  oxygen  and  hydrogen,  the 
constituents  of  water.  It  appeared  from  this  experiment  that 
the  electric  current  can  decompose  water  into  its  constituents. 
When  the  indications  of  this  experiment  were  followed  up,  it 
was  found  that  very  many  compounds  undergo  a  similar  de- 
composition. A  common  characteristic  of  all  the  compounds 
which  undergo  decomposition  was  found  to  be  that  the  com- 
pound must  be  brought  into  the  liquid  state,  either  by  solu- 
tion in  a  solvent  or  by  fusion.  The  products  of  decomposition 
which  are  obtained  are  not  always  the  constituents  of  the 
compound  which  is  dissolved.  In  the  case  just  described,  for 
example,  the  products  obtained  are  not  the  constitutents  of  the 
acid  dissolved  in  the  water,  but  of  the  water  itself.  In  such 
cases  the  direct  action  of  the  current  is  supposed  to  be  com- 
plicated by  secondary  chemical  actions. 

The  quantitative  laws  of  the  chemical  action  of  the  current 
were  discovered  in  1834  by  Faraday. 

187.  The  Electric  Arc. — In  1800,  by  the  use  of  a  powerful 
battery,  Davy  discovered  that  when  the  two  terminal  wires, 
or  better,  when  two  pieces  of  charcoal  or  carbon  which  are 
joined  to  the  terminal  wires,  are  first  touched  together,  so 
that  the  current  is  established,  and  are  then  slightly  separated, 


294  THE   ELECTRIC  CURRENT. 

the  ends  near  the  point  of  contact  become  intensely  heated  and 
brilliantly  luminous,  and  are  connected  by  a  column  of  what 
looks  like  a  flame,  which  is  called  the  electric  arc.  The  light 
in  the  arc  itself  is  generally  bluish  in  color,  and  is  less  intense 
than  that  from  the  carbon  terminals.  When  carbon  rods  are 
used  as  terminals,  a  small  depression,  called  the  crater,  is 
formed  at  the  end  of  the  positive  terminal.  The  highest 
temperature  of  the  arc  is  obtained  in  this  crater.  It  is  esti- 
mated to  be  as  high  as  3400°  Centigrade.  All  known  sub- 
stances except  carbon  can  be  fused  in  the  arc. 

While  the  arc  is  established,  both  the  carbon  rods  waste 
away,  the  positive  rod  wasting  twice  as  fast  as  the  negative 
rod.  This  effect  is  not  due  entirely  to  combustion,  for  the  arc 
may  be  established  in  a  vacuum,  and  the  wasting  away  of  the 
terminals  occurs  in  that  case  also. 

188.  The  Magnetic  Field  of  the  Current. — In  the  year  1820, 
the  Danish  physicist  Oersted,  who  had  been  for  some  time 
hoping  to  find  a  relation  between  the  current  and  magnetism, 
placed  a  wire  carrying  a  current  above  a  compass  needle,  and 
parallel  to  it,  and  noticed  that  the  needle  turned  out  of  the 
magnetic  meridian.  When  the  current  was  reversed,  the 
needle  turned  out  of  the  meridian  in  the  other  sense.  When 
the  wire  was  placed  below  the  needle,  its  deflection  for  the 
same  direction  of  the  current  was  opposite  to  that  obtained 
when  the  wire  was  above  it. 

In  the  same  year  the  action  of  the  current  on  a  magnet  was 
investigated  quantitatively  by  Biot  and  Savart.  In  their  ex- 
periments the  current  traversed  a  long  straight  wire  set  up 
vertically.  By  arranging  magnets  in  the  neighborhood  so  as 
to  neutralize  the  earth's  magnetic  field,  a  region  was  obtained 
around  the  wire  in  which  there  was  no  perceptible  magnetic 
force,  so  long  as  there  was  no  current  in  the  wire.  A  small 
magnet,  suspended  so  as  to  turn  with  freedom  in  any  direc- 
tion, was  used  as  an  indicator  of  the  effect  of  the  current. 
When  the  current  was  set  up,  this  magnet  assumed  a  position 
in  which  it  was  tangent  to  a  circle  drawn  around  the  wire  as 
centre,  in  a  plane  perpendicular  to  it.  When  the  current  was 


THK    ELECTRIC   CURRENT.  296 

in  one  direction,  the  north  pole  of  the  magnet  always  pointed 
in  one  sense  around  this  circle,  at  whatever  point  on  the  cir- 
cle it  was  placed.  When  the  current  was  reversed,  the  direc- 
tion in  which  the  magnet  pointed  was  also  reversed.  We  may 
express  this  result  by  saying  that  the  current  sets  up  and 
maintains  a  magnetic  field,  in  which  the  lines  of  force,  in  the 
case  of  a  long  straight  current,  are  circles  having  the  current 
as  a  common  centre. 

The  direction  of  the  lines  of  force  depends  upon  the  direc- 
tion of  the  current.  We  consider  the  current  as  flowing  from 
higher  to  lower  potential  or  from  the  positive  pole  of  the 
voltaic  cell  through  the  external  circuit  to  the  negative  pole. 
To  express  the  relation  between  the  direction  of  the  current 
and  the  direction  of  the  lines  of  force,  we  may  use  several 
modes  of  statement.  Ampere's  rule  is  that  when  an  observer 
swimming  with  the  current  looks  toward  the  magnet,  the  north 
pole  of  the  magnet  is  deflected  toward  his  left.  Maxwell's 
rule  is  that  the  direction  of  the  current  and  the  direction  of 
the  lines  of  force  are  related  as  the  translation  and  the  rota- 
tion of  a  right-handed  screw.  Another  rule  is  that  if  the 
right  hand  grasps  the  wire,  with  the  thumb  extended  in  the 
direction  of  the  current,  the  fingers  encircle  the  wire  in  the 
direction  of  the  lines  of  force. 

By  allowing  the  indicator  magnet  to  execute  vibrations  in 
the  magnetic  field  of  the  current  at  different  distances  from 
the  wire,  Biot  and  Savart  showed  that  the  magnetic  force  in 
the  field  at  any  point  was  inversely  as  the  distance  of  that 
point  from  the  wire.  Laplace  showed  that  this  law,  which 
expresses  the  integral  effect  of  the  whole  straight  current, 
can  be  obtained  from  the  hypothesis  that  each  element  of  the 
current  acts  on  the  magnet  pole  with  a  force  which  is  inversely 
as  the  square  of  the  distance  between  the  element  and  the  pole, 
and  which  is  also  proportional  to  the  sine  of  the  angle  be- 
tween the  line  joining  the  element  with  the  pole  and  the  direc- 
tion of  the  current  in  the  element. 

The  fact  that  a  current  acts  on  a  magnet,  by  means  of  a 
magnetic  field  which  it  sets  up,  suggested  that  two  currents 


296  THE    ELECTRIC    CURRENT. 

might  perhaps  interact  with  each  other  "by  means  of  their  mag- 
netic fields.  By  experiments  instituted  to  test  this  suggestion, 
Ampere  showed  that  in  fact  currents  do  interact  with  each 
other.  He  furthermore  determined  a  formula  or  law,  ex- 
pressing an  action  between  any  two  elements  of  current,  by 
means  of  which  all  the  forces  exerted  by  currents  on  one  an- 
other can  be  calculated. 

So  long  as  we  confine  our  attention  to  limited  portions  of 
two  circuits  carrying  currents,  and  consider  only  the  forces 
exerted  by  those  portions  on  each  other,  we  may  express  the 
general  mode  of  action  discovered  by  Ampere  by  saying,  that 
parallel  currents  which  are  in  the  same  sense  attract  each 
other  and  those  which  are  in  opposite  sensea  repel  each  other. 
If  the  currents  are  not  parallel,  the  forces  are  such  as  would 
be  exerted  by  parallel  components  of  the  two  currents,  and  are 
therefore  attractions  if  the  components  are  in  the  same  sense, 
and  repulsions  if  they  are  in  opposite  senses. 

When  we  consider  the  action  of  two  complete  circuits  on 
each  other,  or  of  portions  of  two  circuits  so  shaped  as  to  be 
practically  equivalent  to  complete  circuits,  we  are  much 
assisted  by  an  analogy  discovered  by  Ampere  between  the 
magnetic  field  of  a  circuit  and  the  magnetic  field  of  a  so- 
called  magnetic  shell.  The  magnetic  shell  is  an  hypothetical 
form  of  magnet,  consisting  of  a  thin  sheet  of  steel  whose 
faces  are  uniformly  magnetized  positively  and  negatively  re- 
spectively. For  convenience  in  representation  and  description 
we  shall  consider  only  the  plane  magnetic  shell.  On  considera- 
tion of  the  field  of  force  around  such  a  shell,  it  is  plain  that 
the  lines  of  force  which  originate  on  its  north  or  positive  face 
will  be  symmetrical  curves  passing  around  its  outside  edge 
and  terminating  at  points  corresponding  to  the  points  of  origin 
on  its  south  or  negative  face.  When  this  system  of  lines  of 
force  is  compared  with  the  lines  of  force  in  the  magnetic  field 
of  a  current,  travelling  in  a  circuit  which  coincides  with  the 
edge  of  the  shell,  the  two  systems  are  found  to  be  identical 
in  form.  The  direction  of  the  lines  of  force  will  also  be  the 
same  in  both  fields,  if  the  current,  to  an  observer  looking  at 


THE   ELECTRIC   CURRENT.  297 

the  south  face  of  the  shell,  is  travelling  in  the  equivalent 
circuit  in  the  clockwise  direction.  Ampere's  experiments 
proved,  in  effect,  that  the  forces  between  two  closed  circuits 
are  the  same  as  those  between  the  two  magnetic  shells  which 
are  equivalent  to  them.  Now  the  forces  between  two  mag- 
netic shells  can  be  immediately  perceived  from  the  known 
laws  of  magnetic  action,  and  hence  the  forces  between  the 
equivalent  currents  can  be  determined. 

189.  Ampere's   Theory   of   Magnetism. — Ampere   used   the 
principles  which  he  discovered  as  a  basis  for  a  theory  of  mag- 
netism.    In  general   he  adopted  a  theory   similar  to  that  of 
Weber,  in  so  far  as  he  supposed  each  molecule  of  iron  to  be  a 
separate  and  permanent  magnet.     He  then  explained  the  mag- 
netic condition  of  the  iron  molecules  by  supposing  that  each 
of  them  has  an  electric  current  continually  circulating  about 
it.     Such  an  electric  current  would   set  up  a  magnetic   field 
similar  to  that  around  a  very  small  magnet,  and  so  the  mag- 
netism of  the  molecule   is  accounted  for  as  an  electric  phe- 
nomenon.   This  theory  of'magnetism,  or  one  that  is  essentially 
similar   to   it,   though   expressed   in  other   terms,   is   still   the 
most  satisfactory  way  of  explaining  natural  magnetism. 

190.  Electromagnets. — If  a  long  wire  is  coiled  into  a  tight 
spiral,   each   turn   is   practically   a    small   closed   circuit   and 
acts  like  a  magnetic  shell,  so  that  the  magnetic  field  of  the 
whole  spiral  is  like  that  of  a  pile  of  magnetic  shells  arranged 
with  their  similar  faces  in  the  same  direction.     Such  a  pile 
of  magnetic  shells  is  plainly  equivalent  to  a  magnetized  bar, 
and  the  magnetic  field  of  such  a  bar  is  similar  to  that  of  the 
spiral.     The  spiral  used  in  this  way  to  replace  a  magnet  was 
called  by  Faraday  a  solenoid. 

An  important  difference  between  the  field  of  the  solenoid 
and  that  of  the  magnet  must  be  specially  noticed.  The  lines 
of  force  of  a  magnet,  so  far  as  they  can  be  traced  by  experi- 
ment, run  from  the  north  end  of  the  magnet  to  its  south  end. 
On  the  other  hand,  the  lines  of  force  of  the  solenoid  are  closed 
curves,  any  one  of  which  may  be  traced  from  a  point  at  one 
end  of  the  solenoid  through  the  region  outside  it  to  the  other 


20R  THK    ELKCTRIC    CURRKNT. 

end,  in  which  respect  it  is  like  a  line  of  force  of  the  magnet, 
and  then  contimi'S  in  the  mma  sen^e  within  the  solenoid  to 
the  point  of  beginning.  The  lines  of  force  of  the  solenoid 
therefore  form  a  bundle  of  closed  curves.  To  see  whether  there 
is  any  counterpart  in  the  magnet  to  the  lines  of  force  within 
the  solenoid,  it  is  necessary  to  measure  the  magnetic  force 
inside  of  the  magnet.  This  can  only  be  done  by  cutting  cavi- 
ties in  the  magnet  at  different  points  and  by  observing  the 
magnetic  force  within  them.  The  theory  shows  that  the  mag- 
netic force  which  will  thus  be  observed  will  differ  with  the 
shape  of  the  cavity.  If  the  cavity  is  always  a  narrow  crevasse, 
or  disk-shaped  cavity,  with  its  faces  perpendicular  to  the  di- 
rection of  the  force,  the  lines  of  force  determined  by  observa- 
tions within  it  will  be  similar  to  those  within  the  solenoid. 
The  magnetic  force  determined  in  such  a  cavity  is  called  the 
magnetic  induction,  and  the  lines  of  force  thus  determined  are 
called  linos  of  induction.  We  may  therefore  express  the  rela- 
tion between  the  solenoid  and  a  magnet  by  saying,  that  the 
lines  of  force  of  the  solenoid  are  similar  to  the  lines  of  induc- 
tion of  the  equivalent  magnet.  For  a  similar  reason  we  may 
say  in  general  that  the  lines  of  force  of  a  current  are  similar 
to  the  lines  of  induction  of  the  equivalent  magnetic  shell. 

If  a  bar  of  soft  iron  is  placed  within  the  solenoid,  while  it 
carries  a  current,  the  magnetic  field  of  the  solenoid  will  make 
the  bar  a  magnet.  Such  an  arrangement  is  called  an  electro- 
magnet. The  magnetization  of  the  bar  disappears,  or  nearly 
so,  when  the  current  in  the  solenoid  ceases.  Its.  intensity  of 
magnetization  depends  upon  the  quality  of  the  iron,  but 
mainly  upon  the  strength  of  the  magnetic  field  due  to  the 
current,  and  this  depends  on  the  strength  of  the  current  and  on 
the  number  of  turns  made  by  the  circuit  about  its  axis  in  a 
unit  of  length. 

191.  Galvanometers. — Any  instrument  used  to  detect  the 
presence  of  a  current,  or  to  measure  its  strength  by  observa- 
tions of  the  interaction  between  the  current  and  a  magnet,  ia 
called  a  galvanometer.  A  simple  circuit  of  wire  placed  ver- 
tically in  the  plane  of  the  magnetic  meridian,  with  a  magnet 


THK    KLKCTRir   CURRENT.  299 

suspended  in  the  middle  of  it,  will  answer  this  purpose.  The 
portions  of  the  current,  flowing  above  the  magnet  in  one  di- 
rection and  below  it  in  the. opposite  direction,  unite  in  turning 
the  magnet  out  of  the  magnetic  meridian.  The  effect  is  natur- 
ally heightened  when  the  wire  is  turned  on  itself  many  times 
into  a  flat  spiral,  the  strength  of  the  magnetic  field  increas- 
ing with  the  number  of  turns.  In  one  of  the  early  forms  of 
galvanometer,  constructed  by  Schweiger,  a  coil  or  circuit  of 
this  sort  was  used  in  combination  with  an  astatic  needle.  The 
astatic  needle,  or  system,  consists  of  two  similar  light  magnets 
held  rigidly  parallel  to  each  other  by  a  short  connecting  rod. 
These  magnets  are  magnetized  in  opposite  senses,  and  one  of 
them  a  little  more  strongly  than  the  other.  When  such  a  pair 
of  magnets  is  suspended,  the  stronger  one  will  overpower  the 
other,  and  its  north  pole  will  .point  toward  the  north,  but  the 
directive  action  of  the  pair  is  much  feebler  than  that  of  either 
one  of  them.  In  Schweiger's  instrument  the  lower  magnet  of 
the  two  hangs  within  the  coil,  while  the  other  one  is  above 
it.  When  the  current  passes,  its  magnetic  field  turns  both 
these  magnets  in  the  same  sense,  and  since  the  directive  action 
of  the  magnets  in  the  field  of  the  earth  is  very  slight,  the 
deviation  of  the  system  will  be  very  great  in  comparison  with 
that  which  the  same  current  would  produce  in  a  single  mag- 
net. Very  feeble  currents  may  therefore  be  detected  by  this 
instrument. 

A  galvanometer  which  is  of  special  practical  and  theoretical 
importance  is  the  tangent  galvanometer,  so  called  from  one  of 
its  characteristic  properties.  The  coil  of  this  instrument  con- 
sists of  a  number  of  turns  of  wire  wound  in  a  circle,  so  that 
the  thickness  of  the  coil  is  very  small  in  comparison  with^he 
radius  of  the  circle.  This  circle  is  set  up  on  edge  in  the 
plane  of  the  magnetic  meridian,  and  a  short  magnet  is  sus- 
pended at  its  centre.  When  a  current  is  sent  through  the 
coil,  the  magnet  is  turned  out  of  the  meridian,  and  assumes  a 
position  in  which  the  couple  exerted  on  it  by  the  earth's  field 
is  in  equilibrium  with  the  couple  exerted  on  it  by  the  field 
of  the  coil.  The  study  of  the  field  of  the  coil  shows  that  it  is 


300  THK    ELECTRIC   CURRENT. 

of  constant  intensity  in  a  region  lying  around  the  centre  of 
the  coil,  so  that  the  force  exerted  by  the  field  of  the  coil  on  the 
magnet's  pole  does  not  change  with  the  deviation  of  the  mag- 
net. If  we  represent  the  angle  of  deviation  by  0.  the  horizon- 
tal intensity  of  the  earth's  field  by  H,  and  the  strength  of  the 
field  of  the  current  by  R,  and  remember  that  the  lines  of  force 
of  the  current  are  at  right  angles  to  the  plane  of  the  coil  and 
BO  also  at  right  angles  to  the  magnetic  meridian,  it  is  evident 
that  the  couple  exerted  on  the  magnet  by  the  earth's  field  is 
proportional  to  H  f\n<f>  and  the  couple  exerted  by  the  mag- 
netic field  of  the  current  is  proportional  to  R  co£0,  so  that 
we  have  H  sin0  =  R  cos0,  and  the  ratio  of  R  to  H  is  equal 
to  the  tangent  of  the  deviation.  It  is  for  this  reason  that  this 
galvanometer  is  called  the  tangent  galvanometer.  If  we  make 
the  supposition  that  the  strength  of  the  current  in  the  coil  is 
proportional  to  the  strength  of  the  magnetic  field  which  it  sets 
up,  we  may  compare  different  currents  by  comparing  the  tan- 
gents of  the  deviations  which  they  occasion. 

192.  Electromagnetic  Rotations. — A  number  of  arrange- 
ments were  constructed  by  Faraday  in  1821  by  which  the  mag- 
netic force  acting  between  a  current  and  a  magnet  was  made 
to  produce  a  continuous  rotation  of  a  part  of  the  apparatus. 
In  a  typical  one  of  these  instruments  two  shallow  circular 
troughs  are  placed  concentric  with  each  other  around  an  up- 
right post,  on  the  top  of  which  is  fixed  a  bearing.  Two  similar 
magnets  are  placed  vertically  on  opposite  sides  of  the  ap- 
paratus, so  that  similar  poles  stand  each  between  the  troughs, 
and  a  little  above  their  level,  while  the  other  poles  are  con- 
siderably below  that  level.  A  light  wire  frame,  consisting 
of  a  horizontal  portion,  of  one  dependent  portion  which  reaches 
the  inner  trough,  and  of  two  dependent  portions  which  reach 
the  outer  trough,  is  mounted  on  a  rod  whose  end  rests  in  the 
bearing  on  the  top  of  the  post,  so  that  the  frame  can  turn 
about  the  rod  as  a  vertical  axis.  Mercury  is  poured  into  the 
troughs  until  the  ends  of  the  dependent  wires  are  always  in 
contact  with  it  as  the  frame  turns  round  its  axis.  Wires 
from  a  voltaic  battery  introduce  the  current  into  one  trough, 


THE    ELECTRIC    CURRENT.  301 

from  which  it  passes  through  the  frame,  and  so  out  through 
the  other  trough.  When  the  current  is  introduced,  the  frame 
maintains  a  continuous  rotation  about  its  axis.  This  move- 
ment may  be  understood  by  considering  the  force  between  one 
of  the  magnet  poles  and  that  part  of  the  movable  circuit  which 
is  near  it.  If  the  circuit  were  at  rest  and  the  magnet  pole 
were  free  to  move,  it  would  be  attracted  toward  one  face  of 
the  circuit,  would  pass  through  it,  and  then  be  repelled  by 
the  other  face.  That  is,  the  magnet  pole  would  move  as  nearly 
as  possible  along  the  lines  of  magnetic  force  of  the  current. 
Since  the  magnet  is  fixed,  it  is  the  circuit  which  moves  in  the 
opposite  direction  to  that  in  which  the  magnet  would  move 
in  the  case  supposed.  If  the  circuit  were  everywhere  solid,  it 
would  not  be  possible  for  the  magnet  pole  to  pass  through  its 
plane  without  carrying  its  other  pole  with  it,  and  the  effect  of 
the  two  poles  being  opposite,  no  continuous  rotation  would  be 
set  up.  By  making  a  part  of  the  circuit  fluid,  we  have  ar- 
ranged it  so  that  the  solid  part  of  the  circuit  can  pass  from 
one  side  to  the  other  of  one  magnet  pole  without  being  affected 
by  the  opposite  pole. 

The  work  which  is  done  in  sustaining  these  motions  comes 
from  energy  supplied  to  the  circuit  by  the  battery.  As  we 
shall  afterwards  see  more  at  length,  the  circuit  when  in  motion 
is  not  heated  so  much  as  it  is  when  it  is  at  rest.  Ihe  work 
which  is  done  during  the  motion  is  the  equivalent  of  the  heat 
which  has  disappeared. 

193.  Electromagnetic  Induction. — The  production  of  a  mag- 
netic field  by  a  current  suggested  to  many  observers  the  pos- 
sibility of  the  production  of  a  current  by  means  of  a  magnetic 
field.  For  some  years  all  endeavors  to  obtain  such  a  current 
were  unsuccessful.  In  1832  Faraday  succeeded  in  obtaining 
the  expected  action.  In  his  first  experiment  Faraday  used 
two  concentric  spools  of  wire,  one  of  which  could  be  connected 
with  a  voltaic  battery.  This  he  called  the  primary  coil.  The 
other,  called  the  secondary  coil,  was  connected  with  a  sensi- ' 
tive  galvanometer.  With  this  arrangement  Faraday  found 
that,  when  the  current  of  the  battery  was  thrown  into  the 


30^  THK    ELBCTKIC    CURRENT. 

primary  coil,  the  galvanometer  was  deflected  so  as  to  indicate 
a  current  in  the  secondary  con.  This  current  in  the  secondary 
coil  was  only  temporary.  While  the  current  in  the  primary 
coil  was  maintained  unchanged,  the  deflection  of  the  galvano- 
meter ceased  and  no  current  appeared  in  the  secondary  coil. 
When  the  circuit  of  the  primary  coil  was  broken,  so  that  the 
current  in  it  ceased,  a  deflection  in  the  opposite  sense  to  the 
one  observed  before  occurred  in  the  galvanometer,  indicating 
a  current  in  the  opposite  sense  in  the  secondary  coil.  This 
current  was  of  course  also  only  temporary.  The  temporary 
currents  thus  set  up  on  making  and  breaking  the  primary  cir- 
cuit were  called  by  Faraday  induced  currents,  and  they  were 
said  to  be  produced  by  electromagnetic  induction.  By  observa- 
tions of  the  deflections  of  the  galvanometer,  Faraday  proved 
that  the  sense  in  which  the  induced  current  circulates  in  the 
secondary  coil  is  opposite  to  that  of  the  current  in  the  primary 
coil  when  the  primary  circuit  is  made,  and  is  the  samo  as  that  of 
the  current  in  the  primary  coil  when  the  primary  circuit  is  broken. 

Reasoning  that  the  setting  up  of  a  current  in  the  primary 
coil  is,  in  effect,  bringing  the  primary  coil  to  its  position 
near  the  secondary  from  an  infinite  distance,  Faraday  next 
tried  the  experiment  of  moving  the  primary  coil,  while  the 
current  was  established  in  it,  up  to  the  secondary  coil.  He 
found  then  that  the  galvanometer  was  again  deflected,  showing 
a  current  in  the  secondary  coil,  in  the  same  sense  as  that  of 
the  current  produced  when  the  current  was  thrown  into  the 
primary  coil  in  the  first  experiment.  When  the  primary  coil 
was  rapidly  removed  from  the  neighborhood  of  the  secondary 
coil,  a  current  was  produced  in  the  secondary  coil  in  the  oppo- 
site sense. 

Faraday  next  substituted  a  magnet  for  the  primary  coil, 
and  found  that,  when  it  was  moved  up  to  the  secondary  in 
the  proper  way,  an  induced  current  was  developed,  and  that 
an  induced  current  in  the  opposite  sense  was  developed  when 
the  magnet  was  removed.  The  direction  of  the  induced  cur- 
rents produced  by  the  movement  of  a  magnet  may  be  readily 


T11K    ELECTK1C   CURRENT.  303 

obtained  from  the  rule  already  given,  by  remembering  the 
relation  between  the  lines  of  force  of  a  magnet  and  the  direc- 
tion of  the  current  in  the  solenoid  which  is  equivalent  to  the 
magnet. 

Faraday  considered  the  induced  current  to  result  from  the 
change  of  the  magnetic  field  around  the  circuit  in  which  it 
is  produced.  He  looked  on  a  magnet  as  a  body  which  carries 
with  it  a  set  of  lines  of  magnetic  force,  and  he  described  the 
production  of  the  induced  current  by  the  movement  of  a  mag- 
net as  occurring  whenever  the  circuit  in  which  it  arises  cuts 
through  lines  of  force.  It  is  perhaps  easier  for  us  to  con- 
sider the  induced  current  as  produced  by  an  alteration  in  the 
magnetic  field  enclosed  by  the  circuit.  The  induced  current  is 
produced  only  while  the  field  is  changing,  and  ceases  when  the 
field  becomes  constant. 

The  principal  phenomena  of  induced  currents  were  dis- 
covered by  Joseph  Henry  about  the  same  time  that  they  were 
discovered  by  Faraday  and  independently  of  him. 

One  day  after  a  lecture  Faraday  had  his  attention  called 
by  a  gentleman  in  the  audience  to  a  fact  which  he  had  ob- 
served, that  when  a  circuit  is  broken  in  which  a  coil  of  wire 
is  contained,  a  bright  spark  appears  at  the  break.  On  investi- 
gating this  phenomenon,  Faraday  found  that  the  spark  is 
brighter  when  the  wire  in  the  circuit  is  in  the  form  of  a  coil 
than  when  the  same  wire  is  in  the  circuit,  but  is  not  coiled 
up;  and  reflecting  on  this,  he  perceived  that  the  spark  is  due 
to  an  induced  current  set  up  in  the  circuit  itself  by  the 
change  in  its  own  current.  The  action  to  which  this  effect  is 
due  is  called  self-induction.  Self-induction  was  also  discovered 
and  studied  by  Henry.  If  we  apply  the  rules  for  the  produc- 
tion of  the  induced  current  to  a  single  circuit,  considering  it 
to  act  as  both  primary  and  secondary  circuit  at  once,  we  see 
that  when  a  current  is  thrown  into  the  circuit,  an  induced  cur- 
rent will  be  developed  in  the  opposite  sense.  The  effect  of  this 
current  is  to  retard  the  full  development  of  the  primary  cur- 
rent, so  that  the  current  will  be  established  more  slowly  in  a 
circuit  which  is  coiled  up  so  that  its  parts  can  act  inductively 


301  THE    ELECTRIC   CURRENT. 

on  one  another,  than  it  will  be  in  the  same  circuit  if  it  is  not 
coiled.  When  the  circuit  is  broken,  the  departure  of  the  cur- 
rent produces  an  induced  current  in  the  same  sense  as  that  of 
the  original  current,  so  that  a  momentary  current  of  greater 
strength  is  developed.  It  is  to  this  current  that  the  spark  is 
due  which  is  observed  when  a  circuit  is  broken. 

A  general  rule,  by  which  the  direction  of  the  induced  cur- 
rent in  a  circuit  may  be  determined,  was  given  by  Lenz,  and 
is  known  as  Lenz's  law.  If  we  examine  the  direction  of  the 
induced  current  produced  by  bringing  a  primary  current  to- 
ward the  secondary  circuit,  we  notice  that  it  is  in  the  opposite 
direction  to  the  current  which  would  attract  the  primary 
current.  The  induced  current  in  this  case  therefore  tends  to 
repel  the  primary  current,  or  to  oppose  the  work  which  is 
being  done  on  the  primary  circuit.  In  all  other  cases  the 
same  general  rule  holds,  and  we  may  state  Lenz's  law  by  say- 
ing, that  the  sense  of  the  induced  current  is  always  such  that 
the  force  exerted  by  the  current  opposes  the  action  by  which 
it  is  produced. 

An  interesting  experiment  which  exhibits  Lenz's  law  was 
performed  by  Arago,  several  years  before  the  induced  current 
was  discovered.  He  arranged  a  copper  disk  so  that  it  could  be 
rotated  rapidly  around  a  vertical  axis,  and  suspended  above  it 
a  long  magnet.  When  the  disk  was  rotated  the  magnet  was 
deflected  in  the  sense  of  the  rotation,  and  when  suspended  so 
as  to  be  capable  of  free  rotation  around  the  axis  of  the  disk, 
it  rotated  in  the  same  sense  as  that  in  .which  the  disk  was 
rotated.  Arago  could  not  explain  this  experiment,  and  it  re- 
mained unaccounted  for  until  Faraday  discovered  induction. 
It  was  then  explained  by  Faraday  as  a  consequence  of  the 
induction  of  currents  in  the  copper  disk  by  its  movement  in 
the  magnetic  field  of  the  magnet.  According  to  Lenz's  law, 
the  currents  thus  induced  are  all  in  such  a  sense  as  to  oppose 
the  motion  producing  them,  and  hence  there  arises  a  force 
between  the  disk  and  the  poles  of  the  magnet,  tending  to  pre- 
vent the  rotation  of  the  disk,  or  what  amounts  to  the  same 
thing,  to  set  up  a  rotation  of  the  magnet  in  the  sense  of  the 


THK   KLICTRIC   CURBKXT.  805 

rotation  of  the  disk.  That  currents  of  the  sort  assumed  in 
this  explanation  really  exist  in  the  disk  can  be  shown  by 
rotating  a  disk  in  a  strong  magnetic  field,  and  touching  two 
parts  of  it  with  the  terminals  of  wires  leading  to  a  galvano- 
meter. With  this  arrangement  a  continuous  current  will  pass 
through  the  galvanometer  so  long  as  the  disk  is  rotated.  The 
arrangement  is  therefore  a  machine  for  the  production  of  the 
electric  current  by  motions  in  a  magnetic  field.  The  first 
model  of  such  a  machine  was  constructed  by  Faraday. 

The  extensive  use  of  the  electric  current  in  our  modern 
life  originated  in  this  discovery  of  Faraday's,  by  which  it  ia 
made  possible  to  use  mechanical  energy  directly  and  on  a 
large  scale,  for  the  production  of  the  current.  The  so-called 
dynamo-machine,  used  for  this  purpose,  consists  generally  of 
a  large  fixed  electromagnet,  which  furnishes  a  strong  mag- 
netic field,  and  of  a  rotating  portion,  called  the  armature, 
made  of  a  properly  shaped  mass  of  iron,  over  which  wires  are 
wound  in  such  a  manner  that  as  the  armature  is  rapidly 
rotated  by  an  engine  a  current  is  developed  in  them.  By 
means  of  sliding  contacts  between  these  wires  and  the  external 
circuit,  the  current  thus  developed  is  led  off  from  the  ma- 
chine. When  the  current  is  led  into  a  similar  machine  the 
interaction  between  the  magnetic  fields  of  its  various  parts 
sets  up  and  maintains  a  rotation  of  its  armature.  The  ma- 
chine thus  used  is  called  a  motor. 

Other  forms  of  the  dynamo-machine,  which  produce  an 
alternating  instead  of  a  direct  current,  and  other  motors 
which  can  be  operated  by  the  alternating  currents,  have  been 
devised  and  are  now  very  extensively  used. 

The  induction  coil,  called  also  the  Ruhmkorff  coil,  or  the 
inductorium,  is  an  instrument  used  to  produce  induced  cur- 
rents with  high  electromotive  forces.  It  consists  of  a  cylin- 
drical primary  coil  of  coarse  wire,  wound  around  a  central 
core  of  iron,  and  a  secondary  coil,  outside  the  primary  and 
concentric  with  it,  containing  very  many  turns  of  fine  wire. 
An  automatic  circuit  breaker  is  employed,  by  which  the  cur- 
rent from  a  battery  is  rapidly  made  and  broken  in  the  primary 


306  THE    ELECTRIC   CURRENT. 

circuit.  By  the  addition  of  a  condenser  to  the  primary  cir- 
cuit, which  is  charged  when  the  circuit  is  made  and  is  dis- 
charged through  it  when  it  is  broken,  the  development  of  the 
primary  current  is  delayed  and  its  annihilation  is  hastened, 
so  that  the  electromotive  force  of  the  induced  current  in  the 
secondary  coil,  set  up  on  making  the  primary  circuit,  is  de- 
pressed, and  that  of  the  current  set  up  on  breaking  the 
primary  circuit  is  heightened.  The  consequence  is  that  the, 
induced  current  formed  at  the  breaking  of  the  primary  is 
able  to  leap  over  larger  gaps  than  the  other,  and  so,  when  the 
current  from  the  coil  is  sent  through  a  non-conductor,  like  a 
rarefied  gas,  it  acts  almost  as  if  it  were  passing  in  one  direc- 
tion only. 

104.  Thermoelectricity. — In  our  study  of  Volta's  series  it 
was  stated  that  no  current  will  be  set  up  in  a  circuit  formed 
exclusively  of  conductors  of  the  first  class,  or  that,  in  other 
words,  such  a  circuit  will  be  in  electric  equilibrium.  It  was 
discovered  in  1822  by  Seebeck  that  .this  equilibrium  is  dis- 
turbed when  one  of  the  junctions  of  the  conductors  composing 
this  circuit  is  heated.  In  that  case  a  continuous  current  flows 
in  the  circuit,  so  long  as  the  difference  of  temperature  between 
the  heated  junction  and  the  others  continues.  The  current 
thus  formed  is  called  the  thermoelectric  current.  The  strength 
of  the  thermoelectric  current  depends  on  the  difference  of 
temperature  between  the  heated  junction  and  the  others.  It 
also  depends  upon  the  nature  of  the  conductors  at  the  heated 
junction.  These  conductors  are  usually  metals,  and  for  each 
pair  of  metals  the  current  which  is  developed  may  be  accounted 
for  by  supposing  that  a  difference  of  potential  exists  at  the 
point  of  contact  between  the  metals,  and  that  this  difference  of 
potential  changes  with  the  temperature.  This  difference  of 
potential  is  not  that  discovered  by  Volta,  but  very  much  less 
than  that.  The  rate  at  which  this  difference  of  potential 
changes  with  the  temperature  is  called  the  thermoelectric 
power  of  the  pair  of  condurtors,  or  of  the  thermocouple  or 
element.  That  conductor  of  the  couple  from  which  the  thermo- 
electric current  flows  to  the  other,  across  the  heated  junction, 


THK    ELECTRIC   CURRENT.  307 

is  said  to  be  thermoelectrically  positive  to  the  other  conduc- 
tor. Thus,  for  example,  when  a  couple  is  formed  of,  antimony 
and  bismuth,  it  is  found  by  experiment  that  the  current  flows 
from  the  antimony  to  the  bismuth  across  the  heated  junction; 
the  antimony  is  therefore  thermoelectrically  positive  to  the 
bismuth.  Experiment  shows  that  all  the  metals  can  be  ar- 
ranged in  a  Beries,  such  that  each  successive  member  of  the 
series  is  thermoelectrically  positive  to  those  which  tollow  it,  and 
thermoelectrically  negative  to  those  which  precede  it. 

The  current  produced  in  an  ordinary  circuit  by  a  single 
thermoelectric  element  is  very  slight,  but  a  number  of  such 
elements  may  be  joined  in  succession,  so  that  their  differences 
of  potential  are  added  to  each  other,  and  the  current  pro- 
duced by  them  when  their  alternate  junctions  are  heated  may 
be  very  considerable.  -  Such  an  arrangement  of  elements,  called 
a  thermopile,  is  a  very  sensitive  instrument  for  the  detection 
of  radiant  heat. 

It  was  discovered  by  Peltier  in  1834  that  when  a  current 
from  a  battery  or  any  other  source  is  passed  through  the  junc- 
tion of  two  metals  in  the  sense  of  the  current  which  would  be 
set  up  by  heating  that  junction,  the  junction  becomes  cooler. 
If  the  current  is  passed  in  the  opposite  sense,  the  junction  is 
heated.  This  effect  is  known  as  the  Peltier  effect. 

195.  Identity  of  Electricity  from  Different  Sources. — The 
course  of  Volta's  thought  during  the  investigations  which  re- 
sulted in  his  invention  of  the  voltaic  battery  naturally  led 
him  to  believe  that  the  effects  produced  by  the  battery  were 
electric  effects,  and  that  the  battery  was  essentially  equivalent 
to  a  set  of  Leyden  jars  from  which  a  continuous  discharge 
was  passing.  This  supposed  continuous  discharge  in  the  cir- 
cuit of  the  battery  was  called  the  electric  current. 

The  reasons  which  Volta  had  for  the  view  which  has  been 
described  were,  first,  that  a  difference  of  potential  could  be 
shown  to  exist  between  the  two  poles  of  the  battery  similar 
in  kind,  though  generally  very  much  less  in  degree,  to  that 
existing  between  the  two  coatings  of  a  Leyden  jar,  and  second, 
that  both  the  current  and  the  discharge  from  the  Leyden  jar 


308  THK    KLKCTRIC   CURRENT. 

heated  the  conductors  through  which  they  passed.  It  was  felt 
by  Faraday  that  additional  proof  of  this  view  was  desirable. 
He  accordingly  investigated  the  discharge  of  the  Leyden  jar, 
in  order  to  determine  whether  or  not  all  the  effects  which 
are  produced  by  the  current  are  also  produced  by  it.  He 
found  that,  by  proper  arrangements,  the  discharge  of  the  jar 
could  be  made  to  produce  chemical  decomposition.  Tke 
amount  of  chemical  action  was  so  slight  that  it  could  only  be 
detected  by  him  by  a  special  artifice.  He  impregnated  a  piece 
of  blotting  paper  with  a  solution  of  starch  and  of  iodide  of 
potassium,  and  sent  the  discharge  through  it  between  two  ter- 
minal wires  whose  ends  touched  the  paper.  When  the  dis- 
charge had  passed,  small  blue  spots  appeared  around  the  ends 
of  the  wires,  due  to  the  action  upon  the  starch  of  the  iodine 
released  by  the  chemical  action  of  the  discharge.  The  iodine 
was  evolved  at  both  terminals,  though  more  conspicuously  at 
one  than  at  the  other.  This  result  could  not  be  explained  by 
Faraday.  We  now  know  it  to  be  due  to  the  fact  that  the  dis- 
charge of  the  jar  is  oscillatory,  or  undergoes  periodic  changes 
in  direction.  With  our  modern  electric  machines  a  continuous 
chemical  decomposition,  as  for  example,  of  water,  can  be 
maintained.  The  amount  of  this  decomposition,  even  with  the 
most  powerful  machines,  is  extremely  slight  in  comparison 
with  that  produced  by  a  battery. 

Faraday  also  showed,  by  discharging  Leyden  jars  through 
a  circuit  containing  a  galvanometer,  that  the  discharge 
affected  the  magnetic  needle,  or  produced  a  magnetic  field. 
The  same  thing  was  shown  by  Joseph  Henry,  who  magnetized 
sewing  needles  by  placing  them  in  the  axis  of  a  solenoid,  and 
sending  the  discharge  through  it.  In  the  successive  repeti- 
tions of  this  experiment,  Henry  found  that  the  magnetism  of 
the  needles  was  not  always  the  same,  although  the  sense  of 
the  discharge  in  the  solenoid  was  always  the  same.  The  mag- 
net was  formed  with  its  north  pole  sometimes  at  one  end, 
sometimes  at  the  other.  This  remarkable  result  Henry  ac- 
counted for  by  supposing  that  the  discharge  of  the  jar  was  not 
continuous,  but  oscillatory,  the  strength  of  the  current 
diminishing  with  each  oscillation,  and  that  the  magnetic  state 


THE   ELECTRIC   CURRENT.  309 

of  the  needle  was  determined  by  the  last  oscillation  which  was 
strong  enough  to  reverse  the  magnetic  condition  impressed  on 
the  needle  by  the  oscillation  before  it. 

From  these  experiments,  and  many  others  in  which  the 
currents  obtained  from  various  sources  were  compared,,  Fara- 
day concluded  that  electricity  from  all  these  different  sources 
was  identical  in  kind. 

196.  The  Chemical  Relations  of  the  Current.— We  have 
now  passed  in  review  the  principal  facts  which  were  known 
concerning  the  electric  current  up  to  the  year  1834.  It  will 
have  been  noticed  that  in  most  cases  only  qualitative  state- 
ments have  been  made  about  them.  As  the  facts  collected 
became  more  numerous,  and  were  shown  by  Faraday's  ex- 
periments, just  referred  to,  to  be  intimately  connected  with 
one  another,  the  need  of  more  precise  quantitative  knowledge 
of  the  various  relations  of  the  current  came  to  be  felt.  The 
first  domain  in  which  such  knowledge  was  supplied  was  that 
in  which  the  electric  current  is  related  to  chemical  action. 
The  investigation  was  carried  out  by  Faraday,  who  was  him- 
self a  chemist,  as  well  as  a  physicist. 

At  the  outset  of  the  investigation,  starting  with  the  gen-  \ 
eral  knowledge  of  the  chemical  action  of  the  current  which 
has  already  been  described,  Faraday  introduced  a  new  set  of 
terms  to  describe  the  action  under  investigation.  These 
terms  or  names  did  not  commit  him  to  any  theory  of  the 
action.  They  are  simply  descriptive,  and  were  so  well  fitted 
to  their  purpose  that  they  have  always  been  employed  since 
they  were  introduced.  In  this  nomenclature,  the  substance 
which  is  decomposed  by  the  current  is  called  the  electrolyte. 
When  decomposed  it  is  said  to  be  electrolyzed,  and  the  pro- 
cess of  decomposition  is  called  electrolysis.  The  two  terminals 
by  which  the  current  is  introduced  into  the  electrolyte  are 
called  the  electrodes,  the  positive  one,  or  the  one  at  higher 
potential,  being  called  the  anode,  the  other  the  cathode.  The 
products  of  the  electrolysis  are  called  ions,  the  one  which 
appears  at  the  anode  being  called  the  anion,  the  other  the 
cation.  We  shall  use  these  terms  freely  in  our  discussion  of 
this  subject. 


31U  THE    ELECTRIC    CURRKNT. 

Faraday  devoted  special  attention  to  the  electrolysis  of 
water.  He  had  no  means  of  measuring  the  strength  of  the 
current  which  he  used,  but  by  altering  various  features  of  the 
circuit,  in  such  a  way  as  not  to  change  the  current  appre- 
ciably, for  instance,  by  changing  the  size  of  the  electrodes  and 
by  electrolyzing  different  solutions,  he  convinced  himself  that 
the  amount  of  electrolysis  is  independent  of  everything  con- 
nected with  the  circuit  except  the  quantity  of  electricity 
which  passes.  As  we  would  now  put  it,  the  amount  of  electro- 
lysis is  proportional  to  the  total  current,  and  the  rate  at 
which  electrolysis  takes  place  is  proportional  to  the  current- 
strength.  Faraday  generalized  this  conclusion  into  the  law 
which  we  may  state  by  saying,  that  the  amount  by  weight  of 
an  ion  evolved  at  an  electrode  during  the  passage  of  a  current 
for  unit  time  is  proportional  to  the  cur  rent- strength.  After 
galvanometers  were  constructed  by  which  an  independent 
measure  of  current  strength  was  obtained,  this  law  was  in- 
vestigated with  the  utmost  care  and  completely  verified. 

Faraday  also  investigated  the  electrolysis  of  different  com- 
pounds, yielding  different  chemical  elements  as  ions,  by  the 
passage  of  the  same  current.  He  discovered  by  these  experi- 
ments a  most  important  relation.  To  state  it,  it  will  be 
necessary  to  say  a  word  about  the  relations  of  the  chemical 
elements  to  each  other.  In  making  this  statement  we  shall  use 
the  ordinary  atomic  theory. 

From  the  relations  of  the  weights  of  chemical  compounds 
to  each  other,  chemists  have  agreed  that  the  chemical  ele- 
ments exist  in  the  form  of  minute  atoms,  which  combine  with 
one  another  to  form  chemical  compounds;  and  that  the  atom 
of  any  particular  element  has  a  definite  weight,  which  is 
Characteristic  of  that  element.  The  weight  of  the  hydrogen 
Atom  is  usually  taken  as  the  standard  or  unit  weight,  and 
the  weight  of  the  atom  of  another  element  expressed  in  terms 
of  that  standard  weight,  is  called  its  atomic  weight. 

When  chemical  compounds  are  examined  it  is  found  that 
they  generally  consist  of  molecules  containing  a  few  atoms  of 
the  elements  which  form  the  compound.  In  many  instances 
an  element  may  be  removed  from  such  a  compound,  and  an- 


THK    KLECTRIC   CURRENT.  311 

other  substituted  for  it.  When  this  is  done,  it  is  often  found 
that  the  place  vacated  by  one  number  of  atoms  of  one  element 
is  filled  by  a  different  number  of  atoms  of  the  other  element. 
By  such  experiments,  it  may  be  determined  how  many  atoms 
of  one  element  are  equivalent  in  their  combining  power  to  a 
single  atom  of  another  element.  If  we  conceive  such  experi- 
ments to  be  carried  out  by  replacing  hydrogen  atoms  by  a 
single  atom  of  another  element,  the  number  of  hydrogen  atoms 
displaced  by  the  one  atom  of  the  other  element  is  called  the 
valency  of  that  element.  Sometimes  a  small  group  of  atoms 
of  different  sorts,  forming  what  is  called  a  radical,  will  replace 
a  number  of  hydrogen  atoms  in  a  compound.  In  such  a  case, 
the  radical  has  a  valency  equal  to  the  number  of  hydrogen 
atoms  which  it  displaces. 

If  we  suppose  an  element  having  one  valency  to  replace  in 
a  compound  another  having  another  valency,  it  is  plain  that 
the  weights  of  the  two  elements  which  exchange  places  will  be 
proportional  to  their  atomic  weights  divided  by  their  valencies. 
The  ratio  of  the  atomic  weight  to  the  valency  is  called  the 
chemical  equivalent  of  an  element. 

We  are  now  in  a  position  to  state  the  second  law  discovered 
by  Faraday,  by  saying,  that  the  weights  of  the  ions  produced 
at  the  various  electrodes,  when  a  current  passes  through  dif- 
ferent electrolytes,  are  proportional  to  the  chemical  equiva- 
lents of  the  ions.  To  illustrate  this  law  let  us  suppose  that 
the  same  current  is  used  to  electrolyze  water,  sulphate  of 
copper,  and  chloride  of  silver.  The  different  elements  con- 
cerned, which  we  need  to  consider,  are  arranged  in  the  follow- 
ing table,  with  their  respective  atomic  weights,  valencies,  and 
chemical  equivalents: 

Atomic  Chemical 

Elements.  Weight.     Valency.     Equivalent. 

Hydrogen 1  1 

Oxygen  16  2  8 

Copper    63  2  31.5 

Silver    108  1  108 

Chlorine  . .  35.5  1  35.5 


312  THE    ELECTRIC    CURRENT. 

When  the  same  current  is  sent  through  these  compounds  for 
a  while,  and  the  products  of  electrolysis  collected  and  weighed, 
it  will  be  found  that,  for  every  gramme  of  hydrogen,  there 
are  obtained  8  grammes  of  oxygen,  31.5  grammes  of  copper, 
108  grammes  of  silver,  and  35.5  grammes  of  chlorine. 

A  conclusion  can  be  drawn  from  the  facts  embodied  in  the 
second  law  of  electrolysis,  which  gives  us  an  insight  into  the 
nature  of  distribution  of  electricity.  When  we  consider  the 
exact  proportion  between  the  current-strength  and  the  weights 
of  the  ions  which  are  evolved  at  the  electrodes,  we  are  led  to 
consider  the  passage  of  the  current  from  the  electrolyte  to  the 
electrode  as  effected  by  the  transfer  of  electric  charges  to  the 
electrode  by  the  ions  which  are  developed  on  them.  Each 
elementary  ion  which  has  formed  a  part  of  a  molecule  of  the 
compound  brings  to  the  electrode  a  certain  number  of  units  of 
valency.  The  second  law  shows  that,  for  a  given  current,  the 
ions  of  different  compounds  bring  to  their  respective  elec- 
trodes the  same  number  of  units  of  valency.  If,  therefore,  we 
suppose  electricity  to  be  divided  into  equal  portions,  which  we 
may  call  ionic  charges  and  if  we  suppose  that  one  such  ionic 
charge  is  associated  with  each  unit  of  valency,  and  that  the 
charges  of  the  ion  are  surrendered  to  the  electrode  when  the 
ion  is  developed  on  it,  we  then  can  account  for  both  the  laws  of 
electrolysis.  It  is  plain  that  this  conception  is  inconsistent 
with  any  conception  of  electricity  as  being  of  the  nature  of  a 
continuous  fluid. 

197.  Theories  of  Electrolysis. — Within  a  few  years  after 
the  discovery  of  electrolysis,  a  theory  was  advanced  by  Grot- 
thus  to  explain  it.  from  which  the  more  modern  theories  were 
developed.  Grotthus  supposed  that  each  molecule  of  the 
electrolyte  is  made  up  of  two  equally  and  oppositely  charged 
parts,  which  parts  become  the  ions  when  the  molecule  is  broken 
up.  When  a  difference  of  potential  exists  between  the 
electrodes,  so  that  there  is  an  electric  force  in  the  electro- 
lyte, these  molecules  arrange  themselves  in  chains  or  rows, 
with  their  positive  charges  all  pointing  in  one  direction,  toward 
the  negatively  charged  electrode,  or  in  the  direction  of  the  cur- 


THK    ELECTRIC    CURRENT.  313 

rent.  The  molecules  of  the  rows  are  then  broken  into  their 
ions  by  the  electric  force,  and  the  positive  ions  move  toward 
the  cathode,  and  the  negative  toward  the  anode,  thus  setting 
free  an  ion  at  each  end  of  the  row,  and  bringing  the  remaining 
positive  and  negative  ions  together  so  that  they  form  new  mole- 
cule. These  new  molecules  thus  formed  then  arrange  them- 
selves as  before  under  the  electric  force  and  the  process  is 
repeated.  This  theory  or  description  accounts  for  the  evolution 
of  the  ions  at  the  electrodes  and  not  in  the  body  of  the  electro- 
lyte, and  it  indicates  a  relation  between  the  amount  of  electro- 
lysis and  the  strength  of  the  current.  When  Faraday's  laws 
were  discovered,  he  found  that  they  also  could  be  explained  by 
Grotthus'  theory,  although  Faraday  himself  preferred  another 
one,  which  was  not,  however,  essentially  different. 

When  electrolysis  was  studied  more  carefully,  certain  phe- 
nomena were  observed  which  could  not  easily  be  reconciled  with 
this  theory.  In  particular,  it  was  found  that  no  matter  how 
small  the  difference  of  potential  is  between  the  two  electrodes, 
a  current  will  pass  between  them  through  the  electrolyte,  and 
electrolysis  will  occur.  To  account  for  this  and  for  certain 
other  peculiarities  observed,  as,  for  example,  the  fact  that  the 
conductivity  of  an  electrolyte  increases  with  the  temperature,  v 
Williamson  in  England  and  Clausius  in  Germany  modified  the 
theory  of  Grotthus.  In  their  form  of  the  theory  use  was  made 
of  the  hypothesis  that  the  molecules  of  all  bodies  are  in  active 
motion.  This  being  admitted,  it  was  assumed  that  occasionally, 
when  two  molecules  of  the  electrolyte  encounter  each  other, 
they  are  so  affected  by  their  mutual  electric  forces,  or  by  the 
shock  of  impact,  as  to  be  broken  into  their  constituent  ions. 
These  ions  exist  in  the  electrolyte  for  a  time  as  free  ions,  and 
during  that  time  move  toward  the  one  electrode  or  the  other, 
according  as  they  are  positive  or  negative.  This  description  of 
electrolysis  is  no  doubt  a  considerable  advance  on  the  one 
which  we  first  examined. 

A  further  modification  of  the  theory  was  rendered  neces- 
sary by  a  discovery  of  Kohlrausch.  By  determining  the  cur- 
rents in  different  electrolytes  under  similar  conditions,  Kohl- 


314  THK    ELKCTRIC    CURRENT. 

4 

rausch  showed  that  they  can  be  accounted  for  by  supposing 
that  the  ions  of  each  particular  sort,  when  urged  by  a  given 
electric  force,  move  through  the  electrolyte  with  a  velocity 
which  is  peculiar  to  that  sort  of  ion,  and  which  is  the  same 
for  that  ion  from  whatever  combination  it  is  evolved.  Thus,  as 
we  may  say,  the  conductivity  of  an  electrolyte,  that  is,  its 
power  of  conveying  current  under  standard  conditions,  is  the 
sum  of  two  numbers  characteristic  of  its  ions. 

Another  experimental  result  was  also  influential  in  leading 
to  a  modification  of  Clausius'  theory.  It  was  found  that,  dur- 
ing the  process  of  electrolysis,  those  parts  of  the  electrolyte 
near  the  electrode  contain  more  than  the  average  amounts  of 
the  two  ions.  This  movement  of  the  ions  toward  their  re- 
spective electrodes  was  investigated  by  Hittorf.  The  relative 
rates  of  movement  for  the  different  ions  were  found  by  Kohl- 
rausch  to  agree  with  those  which  he  had  determined  from  his 
observations  of  conductivity. 

'  The  requisite  modification  of  the  theory  of  electrolysis,  to 
account  for  these  facts,  was  made  by  Arrhenius.  The  theory 
which  he  developed  is  called  the  dissociation  theory.  Arrhe- 
nius supposes  that  when  a  salt  or  any  other  substance  is  dis- 
solved so  as  to  form  an  electrolyte,  a  certain  proportion  01  »LS 
molecules  separate  into  their  ions,  so  that  a  solution,  whether 
acted  on  by  electric  forces  or  not,  contains  a  large  number  of 
such  free  or  dissociated  ions.  With  each  of  these  ions  there 
are  associated  electric  charges  of  the  proper  sign,  and  propor- 
tional in  quantity  to  the  valency  of  the  ion.  The  effect  of  intro- 
ducing an  electric  force  in  the  electrolyte  is  to  cause  a  drift  of 
these  free  ions  toward  the  electrodes.  When  they  reach  the 
electrodes,  they  give  up  their  charges  to  them  and  are  evolved 
as  the  products  of  electrolysis.  As  the  free  ions  are  suc- 
cessively removed  in  this  way  from  the  electrolyte,  other  mole- 
cules of  the  solute  are  dissociated,  so  that  the  supply  of  free 
ions  is  kept  up.  The  fundamental  assumption  of  this  theory, 
that  a  considerable  proportion  of  the  molecules  of  the  solute 
are  dissociated  in  the  solution  into  two  parts,  is  borne  out  by 
the  effect  which  such  solutes  have  in  lowering  the  freezing 
point  and  in  raising  the  boiling  point  of  their  solutions. 


TlIK    KLKCT.ilC    CURRENT.  315 

198.  The  Voltameter. — Faraday  suggested  the  use  of  the 
electrolysis  of  water  as  a  means  of  measuring  the  quantity  of 
electricity   which   passes  through  a  circuit.     The   instrument 
which  he  proposed  he  called  the  voltameter.     In  one  of  its 
many  forms,  the  voltameter  consists  of  two  graduated  tubes, 
which  stand  vertical  and  are  connected  at  the  bottom  by  a 
short  horizontal  tube.     To  this  connecting  tube  is  joined  an- 
other  through   which   the   superfluous   water   may   flow  away 
which  is  forced  out  as  electrolysis  proceeds.     Platinum  plates 
are   introduced   into  the   two  graduated  tubes  and  joined   to 
wires  carried  through  their  walls.    The  apparatus,  when  ready 
for  use,  is  filled  with  acidulated  water.     When  the  current  to 
be  measured  is  passed  between  the  electrodes,  electrolysis  be- 
gins and  the  two  ions,  oxygen  and  hydrogen,  begin  to  collect 
in  the  upper  parts  of  the  tubes.     The  quantity  of  electricity 
which  passes  while  the  current  is  allowed  to  flow  is  then  de- 
termined by  the  amount  of  oxygen  or  hydrogen  which  is  ob- 
tained, and  if  the  current  is  kept  constant  while  it  is  flowing, 
the  amount  of  either  gas  evolved  in  unit  time  measures  the 
^trength   of  the  current.     In   the  use  of  this   instrument  by 
Faraday,  the  current  indicated  by  it  was  measured  in  terms  of 
an  arbitrary  unit,  namely,  that  current  which  will  evolve  a 
determined  quantity  of  gas  in  a  fixed  time.     In  the  practical 
use  of  the  voltameter  we  still  often  employ  such  a  unit,  con- 
sidering for  example,  that  current  to  be  unit  current  which 
will  evolve  a  cubic  centimetre  of  hydrogen  in  one  minute.    As 
we  now,  however,  know  the  absolute  value  of  the  current  which 
will  produce  this  result,  we  commonly  express  the  strength  of 
a  current  in  absolute  units,  even  when  we  have  used  the  volta- 
meter to  measure  it. 

It  is  obvious  that  any  other  electrolytic  process  may  be 
used  as  a  means  of  measuring  currents.  In  refined  work  it  has 
been  found  best  to  use  the  electrolysis  of  silver  from  a  solution 
of  chloride  of  silver.  The  relation  of  the  absolute  value  of  the 
current  to  the  amount  of  silver  which  it  will  deposit  from  such 
a  solution  has  been  very  carefully  determined. 

199.  The  Voltaic  Cell.— Now  that  we  have  studied  the  pro- 
cess of  electrolysis,  we  are  in  position  to  examine  the  action 


816  THE   ELECTRIC   CURRKNT. 

of  the  voltaic  cell.  As  now  becomes  apparent,  the  conductor 
of  the  second  class  which  is  interposed  between  the  two  plates 
or  poles  of  the  cell,  is  an  electrolyte.  When  the  poles  are 
joined  by  a  conductor  other  than  this  electrolyte,  an  unbal- 
anced diii'erence  of  potential  exists  in  the  circuit,  and  a  cur- 
rent passes  through  the  electrolyte.  This  current  in  the  elec- 
trolyte is  from  the  electropositive  element  of  the  cell  to  the 
electronegative  element,  that  is,  in  the  typical  case  already 
described,  from  the  zinc  to  the  copper,  and  as  it  flows,  the  elec- 
trolyte between  these  elements  is  broken  into  its  ions.  The 
anion  is  evolved  on  the  electropositive  element,  the  cation  on 
the  electronegative  element.  Now  it  always  is  the  case,  in  any 
voltaic  cell,  that  the  anions,  which  reach  the  electropositive 
element,  combine  with  its  atoms  to  form  molecules,  which  are 
dissolved  in  the  electrolyte.  The  electropositive  element,  which 
is  the  negative  pole  of  the  battery,  wastes  away.  The  cation, 
which  is  evolved  at  the  other  element,  will  sometimes  interact 
with  it  and  sometimes  not.  If  it  does  not,  it  appears  as  the 
product  of  electrolysis  at  the  positive  pole.  If  it  does  interact 
with  that  pole,  some  other  substance  is  evolved  by  that  action 
as  a  product  of  electrolysis. 

To  illustrate  this  general  description,  we  may  consider  the 
simple  voltaic  cell  formed  of  plates  of  copper  and  zinc,  im- 
mersed in  a  solution  of  sulphuric  acid.  When  sulphuric  acid 
is  dissolved,  it  dissociates  into  two  positive  hydrogen  ions  and 
a  negative  ion,  called  the  sulphion,  containing  one  atom  of 
sulphur  and  four  of  oxygen.  Each  of  the  hydrogen  ions  is 
univalent,  and  carries  one  positive  ionic  charge.  The  sulphion 
is  bivalent  and  carries  two  negative  ionic  charges.  When  the 
circuit  is  joined,  a  difference  of  potential  exists  between  the 
zinc  and  the  copper  plates,  the  potential  of  the  zinc  being  the 
higher.  The  hydrogen  ions  move  toward  the  copper  plate,  and 
are  evolved  at  it  without  acting  on  it  chemically.  The  sul- 
phions  move  toward  the  zinc  plate,  and  when  they  reach  it 
combine  each  with  an  atom  of  zinc,  so  as  to  form  sulphate  of 
zinc,  which  dissolves  in  the  solution.  Nothing  appears  at  the 
zinc  plate  as  a  product  of  electrolysis,  and  the  plate  gradually 
wastes  away. 


THE   BLKCTBIO   CU11RKNT.  317 

In  giving  this  description  of  the  actions  in  the  cell,  the 
order  of  events  has  been  left  as  uncertain  as  possible.  To  make 
a  definite  statement  about  it  would  involve  deciding  between 
two  rival  theories  of  the  action  of  the  cell,  known  respectively 
as  the  contact  theory  and  the  chemical  theory.  In  the  contact 
theory,  which  was  proposed  by  Volta,  and  which  has  been 
highly  developed  by  Lord  Kelvin,  the  action  of  the  cell  is 
ascribed  to  the  unbalanced  potential  difference  in  the  circuit, 
arising  from  the  contacts  of  the  different  parts  of  the  circuit, 
and  the  chemical  action,  by  which  the  zinc  is  decomposed,  re- 
sults as  a  consequence  of  electrolysis  set  up  by  this  potential 
difference.  In  the  chemical  theory,  held  by  Faraday,  and  advo- 
cated by  Lodge,  the  potential  difference  in  the  circuit  is 
ascribed  to  a  tendency  to  chemical  action  between  the  zinc  and- 
the  sulphions,  by  which  a  condition  of  strain  is  set  up  around 
the  zinc,  and  indeed  throughout  the  cell.  When  the  circuit  is 
joined,  this  strain  can  be  relieved  by  the  flow  of  electricity 
around  the  circuit.  Experiment  has  not  yet  decided  between 
these  two  theories.  It  is  an  argument  in  favor  of  the  chemical 
theory  that  it  places  the  origin  of  the  force  which  urges  the 
current  round  the  circuit  in  the  place  where  the  energy  of  the 
cell  is  being  expended,  that  is.  at  the  surface  of  contact  be- 
tween the  zinc  and  the  acid.  On  the  contact  theory,  the  largest 
part  of  the  potential  difference  by  which  the  current  is  urged 
around  the  circuit,  arises  at  the  contact  between  the  zinc  and 
the  wire  by  which  it  is  joined  to  the  copper,  at  a  place,  there- 
fore, where  no  chemical  action  is  going  on,  and  where  no 
energy  is  being  expended.  Now  in  the  cases  of  the  production 
of  -the  thermoelectric  current  and  of  the  induced  current,  the 
origin  of  the  current  is  evidently  in  those  parts  of  the  circuit 
in  which  energy  is  being  expended,  and  the  analogy  of  these 
instances  inclines  us  to  favor  the  theory  in  which  a  similar 
relation  holds  in  the  circuit  of  the  voltaic  cell. 

When  a  cell  of  the  sort  just  described  is  set  in  operation, 
the  current  developed  by  it  gradually  diminishes  in  strength, 
until  it  becomes  much  weaker  than  it  was  at  first.  This  result 
is  ascribed  to  what  we  call  the  polarization  of  the  cell.  The 


318  THK    ELECTRIC    CURRENT. 

hydrogen  evolved  at  the  copper  plate  collects  on  it  in  a  very 
thin  layer,  and  so,  in  effect,  partially  replaces  the  copper  plate 
by  a  plate  of  hydrogen.  Not  only  is  this  layer  of  hydrogen  a 
very  poor  conductor,  but  also  the  difference  of  "potential  be- 
tween it  and  zinc  is  less  than  that  between  copper  and  zinc. 
For  both  these  reasons  the  current  of  the  cell  is  weakened.  By 
using  platinum  instead  of  copper,  by  roughening  the  surface  of 
the  platinum,  so  that  the  hydrogen  evolved  on  it  more  readily 
forms  bubbles  and  escapes,  and  by  shaking  the  plate,  much  of 
the  hydrogen  can  be  removed  mechanically,  and  a  cell  obtained 
in  which  the  polarization  is  not  great.  Polarization  may  .be 
avoided  also  by  using  other  combinations  of  materials,  so  se- 
lected that  the  ion  which  appears  at  the  positive  pole  is  of  the 
same  sort  as  the  pole  itself.  Thus  in  the  Danlell  cell,  the  ordi- 
nary cell  used  in  telegraphy,  zinc  and  copper  are  used  as  the 
elements,  and  two  liquids  are  used,  a  solution  of  sulphate  of 
zinc  around  the  zinc,  and  a  solution  of  sulphate  of  copper 
around  the  copper.  These  liquids  are  separated  from  each 
other  by  a  porous  jar,  or  by  the  difference  in  their  specific 
gravities.  When  the  current  passes  through  this  cell,  it  evolves 
sulphions  at  the  zinc,  so  that  the  zinc  is  reduced,  and  copper 
ions  on  the  copper,  so  that  the  copper  plate  merely  becomes 
thicker,  but  without  the  character  of  its  surface  changing.  At 
the  surface  where  the  two  solutions  meet,  zinc  ions  proceeding 
toward  the  copper  plate  meet  with  sulphions  proceeding  in  the 
opposite  direction,  and  combine  to  form  sulphate  of  zinc,  so  that 
an  additional  effect  of  the  action  is  to  diminish  the  quantity 
of  sulphate  of  copper  and  increase  the  quantity  of  sulphate  of 
zinc.  In  order  to  prepare  such  a  cell  so  that  it  will  furnish 
current  for  a  long  time  without  charging,  crystals  of  sulphate 
of  copper  are  placed  in  the  solution  of  sulphate  of  copper,  so 
that  the  strength  of  that  solution  may  be  kept  up. 

Other  forms  of  cell  are  used,  in  which  polarization  is 
avoided  by  surrounding  the  positive  pole  with  some  substance 
which  acts  chemically  upon  the  ions  evolved,  and  removes  them 
from  the  pole.  The  Leclanche  cell  and  the  dry  battery,  used  so 
much  for  ringing  electric  bells  and  for  other  work  of  that 


THE    ELECTRIC    CURRENT.  319 

kind,  are  examples  of  such  cells.  In  them  the  depolarizing 
material  works  slowly,  so  that  the  current  cannot  be  run  for 
a  long  time  without  polarizing.  But  when  the  cell  is  used 
only  for  a  short  time,  it  furnishes  a  strong  current,  and  it  ia 
depolarized,  by  the  time  that  it  is  again  needed. 

The  storagfe  >cell,  developed  from  a  cell  constructed  by 
Plante,  contains  two  lead  plates,  coattd  with  oxide  of  lead, 
and  immersed  in  sulphuric  acid.  When  a  current  is  passed 
through  this  cell,  a  principal  part  of  the  action  consists  in  a 
reduction  of  the  oxide  on  one  plate  to  spongy  lead,  and  the 
conversion  of  that  on  ,the  other  plate  into  a  higher  oxide. 
When  the  terminals  of  the  cell  are  joined,  after  it  has  thus 
been  charged,  the  current  which  it  delivers  is  accompanied  by  a 
return  of  the  surfaces  of  the  plates  to  their  initial  condition. 

200.  Ohm's  Law. — In  the  first  quarter  century  in  which  the 
electric  current  was  studied,  the  ideas  concerning  it  lacked 
definiteness,  and  the  terms-  by  which  it  was  described  in  differ- 
ent cases  were  correspondingly  vague.  A  general  distinction 
was  drawn  between  intensity  currents  and  quantity  currents. 
The  intensity  current  was  produced  by  such  an  arrangement 
as  the  voltaic  pile,  or  a  voltaic  battery  containing  a  large  num- 
ber of  cells  in  series,  so  that  the  difference  of  potential  between 
the  two  terminals,  before  the  circuit  was  made,  was  high.  It 
was  found  that  such  a  battery  would  send  through  a  long  wire 
a  current  which  did  not  differ  very  much  from  that  which  it 
would  develop  in  a  shorter  one.  The  quantity  current  was 
developed  by  a  single  cell,  or  by  a  combination  of  cells  so  ar- 
ranged as  to  be  equivalent  to  a  single  one,  in  which  the  plates 
were  very  large.  Such  a  battery  proved,  by  the  amount  of 
electrolytic  action  which  it  would  perform,  that  it  was  sending 
out  a  large  quantity  of  electricity,  but  it  was  not  able  to  send 
any  considerable  current  through  a  long  circuit.  The  reasons 
for  these  differences  and  others  like  them  were  only  vaguely 
understood. 

In  1826  G.  S.  Ohm  published  an  account  of  experiments  on 
different  circuits,  in  which  the  current  was  developed  by  a 
thermoelement  kept  at  a  constant  temperature.  He  measured, 


320  THE   ELECTRIC   CURRKNT. 

by  the  action  of  the  current  on  a  magnet,  the  strengths  of  the 
currents  developed  in  his  circuits  as  their  lengths  and  other 
characteristics  were  altered,  and  felt  himself  justified  in  an- 
nouncing the  law  that  the  current  strength  is  proportional  to 
a  certain  quantity  characteristic  of  the  thermoelement,  or  of 
the  source  of  current  in  the  circuit,  and  inversely  proportional 
to  the  sum  of  two  quantities,  one  of  which  is  a  constant  for 
any  given  source  of  current  and  the  other  varies  with  the  re- 
maining portion  of  the  circuit.  The  sum  of  these  two  quanti- 
ties Ohm  called  the  resistance  of  the  circuit.  The  quantity 
characteristic  of  the  source  of  current  he  called  the  electro- 
motive force.  In  the  following  year,  in  the  absence  of  exact 
methods  of  measurement,  by  which  this  law  could  be  really 
tested,  Ohm  presented  it  as  the  result  of  speculation  on  the 
nature  of  the  electric  current.  It  has  since  been  abundantly 
confirmed  by  the  most  careful  observations. 

Before  proceeding  to  the  discussion  of  Ohm's  law  we  may 
state  it  in  terms  which  will  afterwards  be  defined  by  saying, 
that  in  any  circuit  the  current  equals  the  electromotive  force 
divided  by  the  resistance.  Of  course  it  is  understood,  in 
making  this  statement,  that  the  quantities  in  the  formula  are 
measured  in  a  consistent  system  of  units.  It  should  also  be 
mentioned  that  exceptions  to  Ohm's  law  are  found  in  certain 
circuits  in  which  the  electric  current  is  passing  by  means  of 
the  electric  discharge. 

201.  The  Tangent  Galvanometer. — The  tangent  galvano- 
meter has  been  briefly  described  in  §191.  It  was  invented  by 
Pouillet  in  1837,  and  used  by  him  in  an  investigation  of  Ohm's 
law.  In  the  use  of  the  tangent  galvanometer  to  measure  cur- 
rent, it  is  assumed  that  the  current  is  proportional  to  the  mag- 
netic field  set  up  by  it.  The  strength  of  this  field,  and  there- 
fore the  current  strength,  is  proportional  to  the  tangent  of 
the  deflection.  With  this  instrument  very  exact  measures  of 
the  current  strength  in  different  circuits  may  be  made. 

To  illustrate  the  use  of  thfs  instrument  in  examining  Ohm's 
law,  let  us  suppose  that  we  have  a  circuit  in  which,  as  the 
source  of  current,  there  is  placed  a  thermopile  or  a  battery,  so 


THE   ELECTRIC  CURRENT. 


321 


treated  that  the  conditions  to  which  the  flow  of  current  is  due 
remain  constant.  We  then  say  that  the  electromotive  force  in 
the  circuit  is  constant.  If  the  current  from  this  source  is  led 
through  a  tangent  galvanometer  and  through  an  additional 
conductor,  called  the  external  conductor,  it  will  have  a  certain 
value,  indicated  by  the  tangent  of  the  deflection  of  the  galvano- 
meter needle.  Suppose  the  external  conductor  a  wire  of  a  cer- 
tain length.  If  we  then  lengthen  the  external  conductor  by 
inserting  a  similar  wire  of  the  same  length,  the  current  ia 
diminished.  If  we  insert  another  similar  wire,  of  the  same 
length,  the  current  is  diminished  still  further.  Comparing  the 
strengths  of  the  currents  with  each  other,  we  find  that  they 
may  be  represented  by  a  formula,  in  which  the  numerator  is 
constant,  and  the  denominator  is  the  sum  of  two  terms,  one  of 
which  is  constant,  and  the  other  a  constant  multiplied  by  1,  2, 
or  3,  according  to  the  length  of  the  wire  in  the  circuit.  It 
seems  obvious  that  the  changes  in  the  current  strength  are 
due  to  the  changes  introduced  in  the  circuit  by  the  different 
lengths  of  wire  used.  The  effect  of  the  wire  seems  to  be  pro- 
portional to  its  length.  We  infer  that  the  quantity,  the  value 
of  which  changes  as  the  length  of  wire  changes,  is  character- 
istic of  the  wire,  and  that  the  other  quantity  in  the  denomi- 
nator, which  remains  constant,  is  a  quantity  similar  in  kind 
to  that  related  to  the  wire,  and  is  itself  related  to  the  other 
part  of  the  circuit.  We  call  the  whole  denominator  the  re- 
sistance of  the  circuit,  and  distinguish  the  terms  which  refer 
to  the  source  of  current  and  the  external  circuit  respectively 
as  the  internal  and  the  external  resistance. 

If  the  external  resistance  is  kept  constant  while  the  source 
of  current  is  changed,  and  if  the  difference  of  potential  be- 
tween the  two  ends  of  the  external  resistance  is  measured  by 
an  electrometer,  it  is  then  found  that  the  current  in  the  ex- 
ternal resistance  is  proportional  to  the  difference  of  potential 
between  its  ends.  We  shall  subsequently  see  how  this  observa- 
tion is  generalized  into  the  statement  that  the  current  is  pro- 
portional to  the  electromotive  force. 


322  THK    ELECTRIC   CURRKNT. 

In  describing  the  use  of  the  voltameter,  it  was  stated  that 
currents  are  measured  with  it  on  the  assumption  that  the  cur- 
rent strength  is  proportional  to  the  rate  at  which  electrolysis 
goes  on.  Tn  using  the  tangent  galvanometer  the  current  is 
measured  by  assuming  that  it  is  proportional  to  the  strength 
of  the  magnetic  field  which  it  sets  up.  Experiment  must  be 
employed  to  determine  whether  or  not  these  two  independent 
modes  of  measuring  current  are  consistent  with  each  other. 
This  experiment  consists  in  measuring  currents  in  various  cir- 
cuits by  both  the  voltameter  and  the  tangent  galvanometer. 
It  is  found  by  such  experiments  that  the  two  methods  of  meas- 
uring are  entirely  consistent  with  each  other,  that  is,  the 
magnetic  force  due  to  the  current  and  the  rate  at  which  the 
current  performs  electrolysis  are  exactly  proportional  to  each 
other. 

202.  Absolute  Units. — A  very  important  step  in  advance 
was  taken  by  Weber  in  1856.  not  only  in  electrical  science,  but 
in  the  organization  of  physical  science  in  general,  by  the  in- 
troduction of  the  absolute  system  of  measurement.  This  abso- 
lute system  is  one  in  which  none  of  the  units  are  entirely 
arbitrary,  except  the  three  fundamental  units  of  length,  time, 
and  mass.  All  other  units  are  developed  from  these  by  the  aid 
of  suitable  definitions  based  on  physical  laws.  We  may  illus- 
trate the  definition  of  such  a  unit  by  referring  to  an  example 
which  has  already  been  given.  The  unit  of  force,  the  dyne,  is 
plainly  defined  in  terms  of  these  fundamental  units.  Now  we 
have  defined  the  unit  charge  of  electricity  as  that  quantity 
which  will  repel,  in  vacuum,  an  equal  and  similar  quantity,  at 
the  distance  of  one  centimetre,  with  the  force  of  one  dyne.  In 
this  definition  the  only  quantities  which  enter  are  quantities 
which  are  either  the  fundamental  units  themselves,  or  are  de- 
fined in  terms  of  those  units,  and  the  unit  quantity  of 
electricity  is  therefore  definitely  determined  or  defined  in  terms 
of  the  fundamental  units.  The  system  of  absolute  units  intro- 
duced by  Weber  is  one  in  which  the  various  quantities  which 
occur  in  the  subject  of  electricity,  and  which  are  capable  of 
measurement,  are  defined  in  an  analogous  manner  in  terms  of 


THE    ELECTRIC   CURKKKT.  323 

the  fundamental  units.  This  system  has  peculiar  advantages, 
not  only  in  theoretical  physics,  but  in  the  practical  application 
of  electricity,  in  which  it  is  of  great  importance  to  have  simple 
relations  between  the  electric  units  and  the  units  of  force  and 
energy.  We  shall  proceed  to  examine  some  of  these  units,  and 
illustrate  in  so  doing  the  way  in  which  their  definitions  are 
derived. 

203.  Current. — In  defining  the  unit  current,  we  proceed  on 
the  hypothesis  that  current  is  a  continuous  transfer  of 
electricity.  By  the  definition  considered  in  the  last  section,  we 
have  determined  the  unit  quantity  of  electricity.  We  may 
then,  by  its  aid,  define  the  unit  current,  as  the  current  which 
in  unit  time  will  transmit  unit  quantity  of  electricity  through 
any  cross  section  of  the  conductor.  We  may  interpret  this 
definition  in  terms  of  either  the  one-fluid  or  the  two-fluid 
theory  of  electricity,  by  considering  negative  electricity  flow- 
ing in  one  sense  as  equivalent  to  positive  electricity  flowing  in 
the  opposite  sense.  The  unit  of  current  thus  defined  is  called 
the  electrostatic  unit  of  current. 

Another  absolute  unit  of  current  may  be  defined  from  the 
magnetic  relations  of  the  current.  By  the  aid  of  the  unit 
magnet  pole,  which  has  already  been  defined,  we  may  measure 
the  strength  of  a  magnetic  field,  and  so  define  the  magnetic 
field  of  unit  strength,  as  the  field  in  which  a  force  of  one  dyne 
will  be  exerted  on  a  unit  pole.  Now  a  current  in  the  circum- 
ference of  a  circle  will  exert  a  magnetic  force  on  a  unit  pole 
at  its  centre  which  is  proportional  to  the  circumference  of  the 
circle,  and  inversely  proportional  to  the  square  of  its  radius, 
or,  since  the  circumference  of  the  circle  equals  2wr,  is  propor- 
tional to  — .  We  may  therefore  define  the  unit  current  as 

the  current  which  will  set  up  at  the  centre  of  a  circle  of  unit 
radius,  in  which  it  is  flowing,  a  magnetic  field  whose  strength 
is  2rr.  A  more  usual  definition  «.f  unit  current  is  obtained 
from  the  analogy  between  the  circuit  and  the  magnetic  shell, 
the  two  definitions  being,  of  course,  consistent  with  each  other. 
The  current  thus  defined  is  called  the  electromagnetic  unit  of 
current. 


324  THE    ELECTRIC   CURRENT. 

The  fundamental  units  enter  into  the  definitions  of  the 
electrostatic  unit  and  of  the  electromagnetic  unit  in  different 
ways,  s(b  that  when  the  units  are  compared  with  each  other  by 
measuring  one  of  them  in  terms  of  the  other,  the  ratio  between 
them  is  not  a  simple  number,  but  is  a  number  expressing  a 
velocity.  This  velocity  is  of  the  greatest  theoretical  import- 
ance in  the  theories  of  electricity  and  optics.  It  was  first  de- 
termined by  Weber  and  Kohlrausch.  In  the  method  which 
they  used,  a  condenser  of  known  dimensions  was  charged  to  a 
known  potential,  so  that  the  charge  contnined  by  it  was  known 
in  electrostatic  units.  By  discharging  it  through  a  galvano- 
meter, the  total  current  which  passed  through  the  galvano- 
meter, or  the  quantity  of  electricity  which  passed,  was  meas- 
ured in  electromagnetic  units.  Expressed  as  a  velocity,  uie 
ratio  between  the  two  units  of  quantity  was  found  to  be 
311,000.000  metres  per  second,  the  electromagnetic  unit  being 
the  larger  one.  Many  determinations  of  this  velocity  have 
since  been  made,  by  methods  which  involve  the  comparison  of 
the  units  of  various  electrical  quantities.  The  best  and  most 
recent  results  agree  in  assigning  to  this  velocity  the  value  of 
three  hundred  million  metres  per  second. 

204.  Electromotive  Force. — If  we  still  retain  the  hypothesis 
that  the, current  is  the  transfer  of  electricity,  and  consider  the 
current  flowing  in  a  part  of  the  circuit,  the  ends  of  which  are 
at  the  potentials  V  and  V  in  electrostatic  units,  and  if  in  the 
time  t  the  quantity  of  electricity  Q  in  electrostatic  units  is 
transferred  from  one  end  of  the  conductor  to  the  other,  the 
work  which  is  done  is  equal  to  Q  (V-V),  and  the  rate  at  which 
work  is  done  is  equal  to  this  quantity  divided  by  t.  Now  the 
quantity  of  electricity  transferred,  divided  by  the  time,  meas- 
ures the  current  strength  /  in  electrostatic  units.  The  rate  at 
which  work  is  being  done,  or  energy  is  being  expended,  in  this 
part  of  the  circuit  is  therefore  equal  to  I  (V-V). 

In  a  complete  circuit  in  which  a  current  exists,  energy  is 
being  expended  at  a  certain  rate.  We  assume  that  there  exists 
in  the  circuit  what  we  call  an  electromotive  force,  which  is 
analogous  to  the  difference  of  potential  of  the  last  paragraph, 


THB   ELECTRIC   CTJBBINT.  326 

in  thut  it  conforms  to  an  equation  of  the  same  form.  We 
therefore  define  the  electromotive  force  as  the  power  of  main- 
taining the  expenditure  of  energy  in  the  circuit,  and  express 
its  relation  to  the  energy  and  the  current  by  the  formula 
W=IE,  in  which  IF  represents  the  rate  at  which  energy  is  be- 
ing expended  in  the  circuit,  and  E  the  electromotive  force. 
The  unit  electromotive  force  in  the  electrostatic  system  is 
therefore  the  electromotive  force  in  a  circuit  in  which  the 
electrostatic  unit  of  current  will  expend  one  erg  in  one  second. 

We  measure  electromotive  force  in  the  electromagnetic  sys- 
tem by  the  same  relation.  That  is,  the  energy  expended  in  a 
circuit  is  defined  to  be  equal  to  the  product  of  the  current  and 
the  electromotive  force  measured  in  electromagnetic  units. 
The  unit  electromotive  force  in  the  electromagnetic  system  is 
therefore  the  electromotive  force  in  a  circuit  in  which  the 
electromagnetic  unit  of  current  will  expend  one  erg  in  one 
second. 

205.  Resistance. — In  §200  we  have  seen  that  we  may  set  the 
current  in  a  circuit  proportional  to  the  electromotive  force 
divided  by  a  quantity  which  we  have  called  resistance,  and 
which  is  characteristic  of  the  circuit.  We  may  measure  re- 
sistance in  the  absolute  system  by  means  of  the  relation  ex- 
pressed in  the  equation  IR—E,  from  which,  since  we  already 
know  how  to  measure  the  current  7  and  the  electromotive  force 
E  in  absolute  units,  we  may  measure  the  resistance  R  in  abso- 
lute units.  In  the  equation  as  written,  the  quantities  are  ex- 
pressed in  the  electrostatic  system,  and  from  it  we  see  that  we 
may  define  the  electrostatic  unit  of  resistance  as  the  resist- 
ance of  a  circuit  in  which  the  electrostatic  unit  of  electro- 
motive force  maintains  an  electrostatic  unit  of  current.  Simi- 
larly the  electromagnetic  unit  of  resistance  is  the  resistance 
of  a  circuit  in  which  the  electromagnetic  unit  of  electromotive 
force  maintains  an  electromagnetic  unit  of  current. 

In  connection  with  the  subject  of  resistance  it  may  be 
stated  that  the  resistance  of  a  cylindrical  conductor  like  a 
metallic  wire,  is  found  to  be  proportional  to  the  length  of  the 
wire  and  inversely  proportional  to  its  cross- section.  It  is  also 


326  THE   ELECTRIC  CURRENT. 

proportional  to  the  specific  resistance  of  the  substance,  that  is, 
the  resistance  of  a  cylinder  of  the  same  substance  of  unit 
length  and  of  unit  cross  section. 

206.  Joule's  Law. — The  energy  which  is  introduced  into  the 
circuit  from  the  source  of  the  current  is  expended  in  tne  cir- 
cuit.   In  a  homogeneous  part  of  the  circuit,  as,  for  example,  in 
a  wire  joining  the  poles  of  a  battery,  this  energy  is  converted 
into  heat.     In  1842  Joule  carried  out  a  series  of  experiments 
to  discover  the  relation  between  the  heat  developed  in  such  a 
conductor  and  the  current  in  it.     He  found  that  for  a  given 
conductor  the  heat  developed  is  proportional  to  the  square  of 
the  current,  and  that,  for  a  constant  current,  the  heat  devel- 
oped in  different  conductors  is  proportional  to  the  resistance. 
If  we  measure  current  and  resistance  in  I  he  units  already  de- 
fined, and  measure  heat  in  units  of  energy,  we  may  show  from 
the  equations  of  the  last  two  sections  that  the  heat  developed 
in  unit  time   is   equal   to   the   square   of  the  current   multiplied 
by  the  resistance.     For,  the  heat  developed  equals  the  energy 
expended,  and  this,  from  $204,  equals  I(V-V').     From  Ohm's 
law  applied  to  this  part  of  the  circuit  we  have  IR=V-V\ 
From  these  equations  we  obtain  the  relation  which  has  been 
stated,  and  which  is  known  as  Joule's  law. 

207.  Helmholtz's  Theorem. — When  we  consider  the  whole 
circuit,  it  generally  happens  that  some  of  the  energy  expenued 
in  it  is  used  in  doing  something  else  than  heating  the  con- 
ductor.   It  may,  for  example,  do  chemical  work  in  electrolysis, 
or  heat  the  junction  of  two  metals,  or  lift  a  magnet  into  a  coil. 
In  all  such  cases  as  these,  it  has  been  found  by  experiment 
that  the  rate  at  which  energy  is  expended  is  exactly  propor- 
tional to  the  current.    We  may  therefore  express  the  relation 
between  the  energy  expended  by  the*  source  of  the  current  and 
the  work  that  is  done  throughout  the  circuit  by  the  equation 
IE  =  12R+1A,    in    which    the   factor   A    expresses   the   rate  at 
which  this  extra  work  is  done  by  a  unit  of  current.     From 

this  equation  we  obtain  7=  p —  as  the  expression  for  the  cur- 
rent in  the  circuit.  The  numerator  E-A.  is  obviously  an 


THK    ELECTRIC    CURRENT.  327 

electromotive  force,  and  the  equation  shows  that  when  work 
is  done  in  the  circuit,  in  any  other  way  than  in  heating  the 
circuit,  the  electromotive  force  in  the  circuit  is  less  than  that 
introduced  by  the  source  of  the  current,  by  an  amount  wuiua 
depends  on  the  rate  at  which  the  extra  work  is  being  done.  If 
the  electromotive  force  E  is  suppressed,  leaving  the  conditions 
in  the  circuit  otherwise  the  same,  the  electromotive  force  A 
will  exist  in  the  circuit  in  the  opposite  sense  to  that  of  E,  and 
a  current  will  also  exist  in  the  circuit  in  the  sense  of  the 
electromotive  force  A  so  long  as  the  conditions  are  maintained 
to  which  that  electromotive  force  is  due. 

We  call  the  electromotive  force  A,  which  is  developed  by 
doing  extra  or  local  work  in  a  part  of  the  circuit,  the  counter- 
electromotive  force.  By  this  theorem  we  can  explain  the  vari- 
ous modes  of  producing  the  electric  current.  For  example,  when 
the  current  moves  a  magnet,  a  counter-electromotive  force  is 
set  up  in  the  circuit.  A  similar  movement  of  the  magnet,  due 
to  any  outside  action,  will  cause  the  same  electromotive  force 
to  arise  and  so  will  produce  an  induced  current.  This  relation 
was  first  proved  by  Helmholtz,  who  used  it  in  illustration  of 
the  principle  of  the  conservation  of  energy,  for  it  follows,  as 
we  have  seen,  as  a  consequence  of  that  principle.  Similarly, 
when  the  current  decomposes  an  electrolyte,  and  liberates  dis- 
similar ions,  a  counter-electromotive  force  is  set  up.  If  these 
liberated  ions  are  otherwise  introduced  into  the  electrolyte,  the 
same  electromotive  force  will  arise.  Such  a  combination  is  a 
voltaic  cell.  So  also,  if  a  current  does  work  by  heating  the 
junction  of  two  metals,  a  counter-electromotive  force  is  set  up. 
If  the  same  junction  is  heated  from  an  outside  source,  the  same 
electromotive  force  is  set  up,  and  we  have  the  thermoelectric 
current. 

We  may  illustrate  this  theorem  of  Helmholtz  by  an  experi- 
ment of  Joule's.  Joule  determined  the  amount  of  heat  which 
was  developed  when  a  certain  mass  of  zinc  was  directly  dis- 
solved in  sulphuric  acid.  He  then  determined  the  amount  of 
heat  developed  in  the  circuit  of  a  battery,  while  the  same 
amount  of  zinc  was  consumed  in  the  battery.  He  found  that 


328  THE   ELECTRIC   CURRENT. 

the  same  amount  of  heat  was  developed,  with  this  difference, 
that  it  appeared  throughout  the  circuit  instead  of  appearing 
immediately  at  the  place  where  the  chemical  action  was  going 
on.  He  then  inserted  a  magnetic  motor  in  the  circuit,  which 
was  run  by  the  current,  and  could  be  made  to  do  work  by 
lifting  a  weight.  So  long  as  the  motor  was  at  rest,  or  was 
running  without  doing  work,  the  amount  of  heat  developed  in 
the  circuit,  for  the  consumption  of  the  same  amount  of  zinc, 
was  the  same  as  before.  When  the  motor  did  work,  less  heat 
was  developed.  The  amount  of  heat  which  did  not  appear  was 
proportional  to  the  work  done  by  the  motor,  and  measured  in 
terms  of  energy  units  was  equal  to  it. 

208.  Practical  Units. — The  direct  measurement  of  current, 
electromotive  force,  and  resistance,  in  their  absolute  units,  is 
extremely  difficult  and  in  ordinary  circumstances  impracti- 
cable. It  has  been  found  necessary  to  establish  intermediate 
standards,  which  are  more  easily  used  in  ordinary  measure- 
ments, and  to  determine  the  values  of  those  standards,  once 
for  all,  in  absolute  units. 

If  we  know  the  value  of  the  horizontal  intensity  of  the 
earth's  magnetism,  we  can  compare  with  it  the  magnetic  force 
set  up  by  a  current  in  the  tangent  galvanometer,  and  can  cal- 
culate the  value  of  that  current  in  electromagnetic  units,  if  we 
know  the  dimensions  of  the  galvanometer.  The  absolute  value 
of  a  current  is  often  determined  in  this  way  in  ordinary  labora- 
tory practice,  but  unless  the  galvanometer  is  very  exactly 
made,  and  the  operation  with  it  conducted  with  extreme  care, 
the  value  obtained  cannot  be  depended  on  as  accurate.  Accord- 
ingly, several  observers  have  compared  the  strength  of  a  cur- 
rent, measured  with  a  very  exact  tangent  galvanometer,  with 
the  amount  of  silver  which  the  same  current  deposits  in  a 
second.  They  thus  found  that  the  amount  of  silver  deposited 
in  one  second  by  the  electromagnetic  unit  of  current  is  11.175 
milligrams.  By  the  aid  of  this  number  a  current  which  is 
measured  directly  by  the  silver  which  it  deposits  can  be  ex- 
pressed in  absolute  units. 


THK    ELECTRIC    CUKRENT.  329 

The  electromagnetic  unit  of  current  is  not  the  one  which  is 
used  in  ordinary  practice.  The  practical  unit  of  current,  called 
the  ampere,  is  equal  to  lO"1  absolute  units  of  current. 

To  determine  the  absolute  value  of  a  resistance  we  must 
determine  in  absolute  units  the  current  and  the  electromotive 
force  in  the  circuit.  The  method  employed  by  the  Committee 
of  the  British  Association  on  Electrical  Standards  will  serve  «?s 
an  illustration  of  how  this  may  be  done.  The  wire  whose  re- 
sistance was  to  be  determined  was  wound  on  the  circumference 
of  a  large  circle  into  a  coil  whose  ends  were  joined  to  each 
other.  This  coil  was  mounted  so  that  it  could  rotate  at  a  de- 
terminate rate  around  its  vertical  diameter.  A  small  magnetic 
needle  was  hung  at  the  centre  of  the  coil.  When  the  coil  was 
turned,  it  cut  through  the  lines  of  force  of  the  earth's  magnetic 
field,  and  so  set  up  induced  currents  in  itself.  The  electro- 
motive force  in  the  coil  was  calculated  from  the  rate  of  change 
of  the  number  of  lines  of  force  encircled  by  the  coil.  The  in- 
duced currents  in  the  coil  all  tended  to  turn  the  magnet  in  the 
same  sense,  and  the  strength  of  the  magnetic  field  produced  by 
them,  and  so  the  strength  of  the  current  in  the  coil,  was  de- 
termined from  the  deflection  of  the  magnet.  When  the  electro- 
motive force  and  the  current  were  both  known  in  absolute 
units,  the  ratio  between  them  gave  the  resistance  of  the  coil  in 
absolute  units.  By  comparison  with  the  resistance  of  this  coil, 
a  wire  of  known  resistance  was  constructed  as  a  standard,  and 
by  comparison  with  it  standard  sets  of  resistances  have  been 
made,  with  which  other  resistances  can  be  compared. 

The  electromagnetic  unit  of  resistance  thus  determined  is 
too  small  to  be  of  practical  use.  Instead  of  it  a  practical  unit 
is  used,  called  the  ohm,  which  was  originally  designed  to  be 
equal  to  TO9  electromagnetic  units  of  ivsisUinco.  It  b;i-  l>een 
found  more  convenient  to  define  the  ohm  as  the  resistance  of  a 
column  of  mercury,  one  millimetre  in  cross-section  and  106.3 
centimetres  in  length,  at  the  temperature  of  melting  ice.  The 
resistance  of  such  a  column  is  very  nearly  equal  to  the  ohm  as 
previously  defined. 


330  THE    ELECTRIC    CURRENT. 

The  electromotive  force  of  a  circuit  may  be  measured  by 
measuring  the  current  and  the  resistance  of  the  circuit  in  abso- 
lute units.  Certain  voltaic  cells  have  been  constructed  whose 
electromotive  force  is  very  constant,  and  reproduced  with  great 
precision  when  the  cells  are  made  up  according  to  a  prescribed 
formula.  The  electromotive  forces  of  these  cells  have  been 
very  carefully  determined,  so  that  they  serve  as  intermediate 
standards. 

The  electromagnetic  unit  of  electromotive  force  is  too  small 
to  be  of  use  in  practice.  We  use  instead  of  it  a  practical  unit, 
called  the  volt,  which  is  equal  to  108  electromngnetic  units  of 
electromotive  force.  The  electromotive  force  of  the  Daniell's 
cell  is  a  little  greater  than  one  volt. 

The  energy  expended  in  a  circuit  in  one  second  is  measured 
by  the  product  of  the  current  and  the  electromotive  force  in 
the  circuit.  When  the  current  is  the  ampere  and  the  electro- 
motive force  is  the  volt,  me  energy  expended  in  one  second,  or 
the  rate  at  which  energy  is  expended,  is  taken  as  a  unit  rate  of 
expenditure  of  energy.  This  unit  is  called  the  watt.  It  is 
equal  t<>  TO7  ergs  per  second. 

209.  Theories  of  the  Electric  Current. — From  the  first  the 
electric  current  was  thought  of  as  a  continuous  transfer  of 
electricity  around  the  circuit.  In  the  one-fluid  theory  this 
transfer  was  all  in  one  direction;  in  the  two-fluid  theory  posi- 
tive electricity  was  supposed  to  move  in  one  direction,  and  an 
equal  amount  of  negative  electricity  in  the  opposite  direction. 
It  was  plain  that  the  interactions  of  currents  could  not  be 
accounted  for  by  the  known  laws  of  electric  attraction  and 
repulsion,  but  no  one,  not  even  Ampere,  ventured  to  suggest 
any  modification  of  those  laws  by  which  the  actions  of  cur- 
rents could  be  explained.  The  question  was  finally  taken  up 
by  Weber,  who  undertook  to  explain  the  actions  of  currents 
by  the  hypothesis  that  moving  charges  act  on  each  other,  not 
only  with  their  electrostatic  forces,  but  also  with  other  forces 
which  depend  upon  their  velocities.  By  developing  this 
hypothesis,  Weber  was  able  to  account  for  the  actions  of  steady 
currents  on  each  other.  By  making  the  further  hypothesis 


THK    ELECTRIC    CURRENT.  331 

that  additional  forces  arise  when  the  velocities  of  the  moving 
charges  are  changing,  Weber  was  also  able  to  account  for  the 
induction  of  currents. 

The  theory  of  Weber  assumed  that  the  actions  between 
currents  take  place  in  some  direct  manner,  which  does 
not  depend  in  any  way  on  the  medium  between  the  currents, 
and  which  depends  only  on  the  distance  between  them.  It  may 
therefore  be  called  a  theory  of  electric  action  at  a  distance. 
The  researches  in  electricity  and  magnetism  in  which  Faraday 
was  for  many  years  engaged  convinced  him  that  electric  and 
magnetic  actions  are  not  simply  actions  at  a  distance,  but  take 
place  by  means  of  the  intermediate  actions  of  some  medium. 
By  reflecting  on  these  views  of  Faraday,  Maxwell  was  led  to 
attempt  to  give  them  mathematical  form,  and  so  to  construct  a 
theory  of  what  we  may  call  medium-action.  One  form  of  his 
theory,  in  which  he  describes  a  medium  which  will  account  for 
electrostatic  action,  has  already  been  described.  One  feature 
of  this  theory,  and  indeed  of  Maxwell's  theory  in  all  its  forms, 
is  that  there  can  be  no  such  thing  as  an  open  circuit,  that  is, 
there  can  be  no  flow  of  electricity  which  begins  and  ends  in  a 
limited  conductor.  According  to  his  theory,  the  flow  of  elec- 
tricity in  the  conductor  is  accompanied  by  an  electric  displace- 
ment in  the  dielectric  around  the  conductor,  so  that  the  cir- 
cuit is  completed  through  the  dielectric.  While  this  displace- 
ment is  being  set  up  in  the  dielectric,  it  is  so  far  like  a  cur- 
rent that  it  produces  the  same  magnetic  field  that  a  current 
of  the  same  strength  and  similarly  distributed  would  produce. 

In  the  most  general  form  of  Maxwell's  theory,  the  hypo- 
thesis is  made  that  the  ether  and  the  electricity,  which  is  dis- 
tributed everywhere  in  it  and  which  can  be  displaced  in  a  di- 
electric and  move  freely  in  a  conductor,  conform  to  the  general 
principles  of  mechanics,  in  so  far  that  if,  to  the  quantities 
which  enter  into  the  general  equations  of  dynamics,  there  be 
given  an  electric  interpretation,  those  equations  will  represent 
the  mode  of  action  of  the  electric  and  magnetic  fields.  By  the 
development  of  this  hypothesis  Maxwell  was  able  to  show  that 
it  leads  first  to  the  induction  of  currents,  then  to  the  actions 


dbli  THE    ELECTRIC    CUKRENT. 

of  steady  currents,  and  finally  to  the  actions  of  electricity  in 
equilibrium. 

If  we  adopt  Maxwell's  theory,  we  conceive  the  act  of  set- 
ting up  a  current  to  involve  not  only  starting  a  stream  of 
electricity  along  the  conductor,  but  also  setting  in  action — we 
might  by  analogy  say,  in  motion — the  etherial  mechanism  in 
the  conductor  and  in  the  dielectric  surrounding  the  conductor. 

When  this  mechanism  is  in  action,  the  region  occupied  by 
it  becomes  the  magnetic  field  of  the  conductor.  To  set  up  this 
field  requires  energy,  and  since  the  energy  from  the  source  is 
supplied  only  at  a  certain  rate,  and  is  at  first  divided  be- 
tween that  expended  in  the  circuit  and  that  spent  in  setting 
up  the  magnetic  field,  the  current  in  the  conductor  rises  only 
gradually  to  its  full  value.  When  it  is  fully  established  the 
maintenance  of  the  action  in  the  magnetic  field  requires  no 
more  energy.  According  to  the  mode  of  representation  adopted 
by  Poynting,  the  flow  of  energy  from  the  source  into  the  di- 
electric does  not  stop  when  this  steady  state  is  reached,  but 
is  continued  at  a  constant  rate,  and  the  energy,  passing 
through  the  dielectric  in  a  certain  definite  manner,  reaches 
the  conductor  by  way  of  the  dielectric  and  is  transformed  in  it. 
When  the  circuit  is  broken,  and  the  flow  of  electricity  ceases, 
the  conditions  which  sustain  the  action  of  the  mechanism 
cease  also,  and  the  energy  stored  up  in  the  dielectric  leaves 
it  and  appears  in  the  conductor  as  the  energy  of  the  extra 
current. 

When  the  current  is  steady,  Poynting's  theorem  shows  that 
tho  expenditure  of  oneriry  in  an  ordinary  homogeneous  part  of 
the  circuit,  resulting  in  the  development  of  hent  in  it,  will  be 
uniform  across  the  whole  cross-section  of  the  conductor.  But 
this  will  not  be  the  case  when  the  current  varies  periodically. 
A  current  which  so  varies  may  be  developed  by  a  properly 
constructed  dynamo-machine,  which  is  arranged  to  set  up  an 
electromotive  force  in  the  circuit  varying  continuously  and 
periodically  between  two  extreme  and  oppositely  directed 
values.  Such  a  current  is  called  an  alternating  current.  It 
was  shown  by  Heaviside  that  the  energy  which  enters  the  con- 


THE   ELECTRIC  CURRENT.  333 

ductor  from  the  dielectric,  when  the  current  in  it  is  alter- 
nating, is  transformed  into  heat  before  the  current  distributes 
itself  uniformly  over  the  whole  cross-section  of  the  conductor, 
so  that  the  principal  development  of  heat  is  in  the  outer 
layers  of  the  conductor.  This  effect  has  been  demonstrated  by 
experiment.  The  extent  to  which  the  current  penetrates  the 
conductor  depends  on  the  period  of  the  alternations,  being 
greater  as  this  is  longer.  For  very  rapid  alternations,  the  cur- 
rent is  confined  almost  entirely  to  the  surface  of  the  conductor. 

This  action  of  alternating  currents  is  a  consequence  of 
Maxwell's  theory,  and  may  be  considered  a  verification  of  it. 

Still,  Maxwell's  theory,  if  it  had  gone  no  further  than  this, 
would  have  been  little  more  than  a  simple  alternative  to  the 
theory  of  Weber,  to  be  adopted  by  those  who  prefer  to  think 
in  terms  of  medium-action  rather  than  in  terms  of  action  at  a 
distance. 

Maxwell,  however,  was  able  to  draw  from  this  theory  an- 
other more  important  series  of  consequences,  which  do  not  fol- 
low from  Weber's  theory,  and  which  were  subsequently  veri 
fied  by  experiment.  We  shall  now  turn  our  attention  to  this 
portion  of  Maxwell's  theory  and  to  its  experimental  demon- 
stration. 

210.  Electromagnetic  Waves. — It  has  already  been  de- 
scribed how  the  flow  of  electricity  in  a  limited  conductor,  on 
Maxwell's  theory,  rs  accompanied  by  an  electric  displacement 
in  the  dielectric.  If  this  flow  alternates  in  direction,  or  ia 
oscillatory,  the  displacement  in  the  dielectric  will  undergo 
similar  oscillations.  It  follows  from  Maxwell's  theory  that 
when  such  oscillations  are  set  up,  they  will  proceed  outward 
from  their  origin,  as  electromagnetic  waves,  with  a  definite 
velocity.  In  the  ether,  according  to  the  theory,  this  velocity  is 
that  which  expresses  the  ratio  between  the  electrostatic  and 
the  electromagnetic  units.  In  other  bodies  it  is  equal  to  this 
velocity  divided  by  the  square  roots  of  their  dielectric  con- 
stants. In  all  their  essential  characteristics  these  waves  are 
exactly  like  light  waves.  In  default  of  direct  experimental 
evidence,  Maxwell  ventured  to  assume  that  the^waves  of  light 


334  THK    ELtCTKIC    CL'HKENT. 

are  electromagnetic  waves,  and  to  examine  their  properties  for 
a  confirmation  of  his  theory.  A  comparison  of  the  velocity 
which  is  the  ratio  between  the  two  systems  of  units  with  the 
velocity  of  light  showed  that  the  two  velocities  are  very  nearly 
equal.  More  accurate  determinations  of  both  these  quantities 
have  shown  that  they  are  really  equal  within  the  limits  of 
probable  error.  The  velocity  of  light  in  any  other  body  than 
ether  is  equal  to  its  velocity  in  the  ether  divided  by  its  index 
of  refraction.  If  therefore  light  waves  are  electromagnetic 
waves,  the  index  of  refraction  of  a  substance  ought  to  be  equal 
to  the  square  root  of  its  dielectric  constant.  In  the  case  of 
paraffine,  which  was  the  only  body  for  which  he  knew  both  the 
index  of  refraction  and  the  dielectric  constant,  Maxwell  found 
that  this  relation  nearly,  though  not  exactly,  held  true.  Later 
investigation  has  shown  that  it  rarely  holds  true  with  light 
waves,  but  an  explanation  of  this  circumstance  can  be  given 
which  permits  us  still  to  regard  light  waves  as  electromagnetic 
waves.  In  addition,  according  to  the  theory,  good  conductors 
ought  to  be  opaque  to  light  and  poor  conductors  transparent, 
for  when  the  electric  disturbance  in  the  wave  falls  upon  a 
good  conductor,  it  should  set  up  a  current  and  its  energy 
should  be  transformed  into  heat,  while  in  a  poor  conductor  it 
should  continue  without  any  such  transformation.  Maxwell 
found,  in  fact,  that  this  relation  holds  generally  true,  although 
the  degree  of  transparency  exhibited  by  the  conducting  metals 
when  in  thin  sheets  was  greater  than  he  had  expected.  Later 
observation,  and  an  extension  of  the  theory,  have  shown  that 
the  facts  of  observation  confirm  the  theory. 

Maxwell  was  able  to  go  no  further  than  this,  and  the 
electromagnetic  theory  of  light,  as  it  was  called,  remained  an 
interesting  speculation,  with  some  evidence  in  its  favor,  umtil 
the  year  1888,  when  the  study  by  Hertz  of  electromagnetic 
waves  and  their  properties  confirmed  the  general  theory  of 
Maxwell  in  the  most  complete  manner. 

To  set  up  electromagnetic  waves,  Hertz  used  the  electric 
spark  passing  between  two  knobs  joined  by  short  rods  to  two 
similar  metals  plates  standing  in  the  same  plane.  This  ar- 


THhi    KLKCTKIC    C'UKKKXT.  33-3 

rangement  is  called  the  vibrator.  As  was  shown  by  the  ex- 
periments of  Faraday  and  Henry,  and  confirmed  by  the  cal- 
culations of  Lord  Kelvin  and  the  observations  of  Feddersen, 
the  discharge  in  such  an  electric  spark  is  oscillatory.  It  is  not 
the  passage  of  electricity  in  one  single  leap  across  the  gap, 
but  is  instead  a  succession  of  passages  of  electricity,  alter- 
nately in  opposite  directions  and  in  diminishing  quantity, 
until  equilibrium  is  attained.  These  alternate  passages  occur 
at  equal  intervals  of  time,  and  are  therefore  adapted  to  serve 
as  the  origin  of  a  short  train  of  waves. 

Hertz  detected  the  passage  of  these  waves  by  using  a  wire 
rectangle  or  circle,  in  which  a  small  gap  was  opened  so  that, 
if  a  current  was  set  up  in  the  circuit,  its  presence  could  be 
detected  by  a  spark  at  the  gap.  He  constructed  this  circuit  of 
such  a  length  that  the  period  of  the  free  oscillation  of  elec- 
tricity in  it  was  the  same  as  that  of  the  discharge.  The 
electric  impulses  coming  to  it  in  the  wave  were  therefore  syn- 
chronous with  its  own  free  vibration,  and  thus  the  electric 
effects  in  it  were  heightened.  This  instrument  was  called  the 
resonator.  After  showing  that  he  could  detect  an  electric  dis- 
turbance, which  was  presumably  an  electric  wave,  whenever  the 
spark  passed  at  the  vibrator,  Hertz  used  the  resonator  to  com- 
pare the  velocity  of  electric  waves  in  a  wire  with  their  velocity 
in  air.  The  results  of  these  experiments  were  Inconsistent  with 
Maxwell's  theory,  but  Hertz  afterwards  recognized  that  there 
was  probably  some  undiscovered  error  in  the  way  in  which  the 
experiments  were  made,  and  the  observations  of  others  who 
have  tried  similar  experiments  confirm  the  conclusions  of  the 
theory,  that  the  velocity  of  the  electromagnetic  waves  In  the 
wire  is  the  same  as  that  in  the  air. 

According  to  the  theory  an  electromagnetic  wave  is  re- 
flected with  change  of  sign  at  a  conducting  surface  on  which 
it  falls  perpendicularly.  A  node  should  therefore  be  developed 
at  that  surface,  and,  if  the  reflected  wave  is  sufficiently  in- 
tense, a  standing  wave,  or  a  succession  of  nodes  and  ventral 
segments,  ought  to  be  set  up.  By  using  a  large  sheet  of  zinc 
as  a  reflector  Hertz  was  able  to  detect  several  of  these  nodes, 


336  THE   ELECTRIC   CURRENT. 

and  so  to  measure  the  length  of  the  electromagnetic  wave. 
Knowing  the  period  of  the  wave  by  calculating  it  from  the 
dimensions  of  the  vibrator,  he  was  able  to  determine  its 
velocity.  He  found  it  to  be  of  the  same  order  of  magnitude  as 
the  velocity  of  light.  Subsequent  observations  have  proved 
that  the  velocity  of  light  and  the  velocity  of  the  electro- 
magnetic waves  are  the  same. 

Hertz  constructed  a  large  prism  of  pitch  and  with  it  found 
that  the  electromagnetic  waves  are  refracted  like  light  waves. 
The  index  of  refraction  which  he  determined  was  of  the  same 
order  of  magnitude  as  that  obtained  for  similar  substances 
with  light. 

The  disturbances  in  the  spark  at  which  the  electromagnetic 
waves  originate  are  all  in  one  line,  and  so  the  waves  which 
come  out  from  the  origin  are  polarized.  Hertz  proved  this  to 
be  the  case  by  interposing  in  the  path  of  the  wave  a  screen 
made  of  a  large  number  of  parallel  wires  set  at  small  dis- 
tances from  each  other.  When  these  wires  were  set  parallel 
with  the  direction  of  the  spark  they  acted  as  a  conducting 
surface  to  absorb  or  reflect  the  vibrations.  When  they  were  set 
perpendicularly  to  the  vibrations,  their  effect  was  not  nearly 
so  marked,  and  the  waves  passed  through  the  screen.  The 
polarization  of  the  electromagnetic  waves,  and  the  complete 
analogy  between  their  behavior  and  that  of  light  waves,  was 
shown  by  the  experiments  of  Trouton.  Trouton  allowed 
electromagnetic  waves  to  fall  obliquely  upon  a  thick  stone 
wall,  and  observed  the  reflected  and  refracted  waves.  He 
found  in  general  that  the  relative  intensities  of  the  reflected 
and  refracted  waves  depended  upon  the  angle  of  incidence,  and 
also  upon  the  angle  between  the  direction  of  the  electric  vibra- 
tion and  the  plane  of  incidence  and  that  in  particular,  when  the  elec- 
tric vibration  was  in  the  plane  of  incidence,  and  the  an^le  of  inci- 
dence had  a  certain  value,  corresponding  to  the  polarizing  angle 
of  light,  the  reflected  wave  entirely  disappeared.  In  fact,  the 
behavior  of  these  waves  was  exactly  like  that  of  polarized  light 
under  the  same  conditions. 

Numerous  experiments  have  been  made  with  electromag-- 
netic  waves  to  test  Maxwell's  relation  between  the  dielectric 


THK   ELECTRIC  CURRENT.  887 

constant  of  a  substance  and  its  index  of  refraction.  The  theory 
has  been  fully  confirmed.  The  reason  why  light  wares  do  not 
generally  show  an  agreement  with  the  theory  seems  to  be  that 
the  dielectric  constant  is  not  the  same  for  the  rapidly  alter- 
nating electromotive  forces  of  the  light  waves  as  it  is  for  the 
slower  alternations  by  which  alone  we  can  determine  it. 

By  these  and  many  similar  experiments  the  properties  of 
the  electromagnetic  waves  have  been  shown  to  agree  precisely 
with  those  deduced  from  the  theory  of  Maxwell.  We  may 
therefore  consider  this  theory  as  confirmed,  at  least  in  its 
essential  features.  We  may  consider  it  proved  that  electrical 
action  takes  place  in  a  universal  medium,  which  is  essentially 
the  same  everywhere,  except  in  so  far  as  its  properties  are 
modified  in  particular  places  by  the  material  bodies  there 
present.  From  the  general  agreement  in  behavior  between 
electromagnetic  waves  and  light  waves,  it  may  also  be  taken 
as  proved  that  this  medium  is  the  ether,  with  the  properties 
of  which  we  have  to  some  extent  become  acquainted  in  our 
study  of  light.  We  may  therefore  adopt  the  electromagnetic 
theory  of  light  without  reserve,  and  consider  the  subjects  of 
magnetism,  electricity,  and  light,  as  forming  parts  of  a  gen- 
eral science  of  the  ether. 

211.  Pressure  of  Light. — A  remarkable  conclusion  of  the 
electromagnetic  theory  is  that  light  will  exert  a  pressure 
against  a  surface  upon  which  it  falls.  This  conclusion  was 
unconfirmed  for  many  years.  An  indirect  proof  of  it  was  given 
by  Boltzmann,  who  showed  that  if  it  were  correct,  it  could  be 
proved,  by  the  methods  of  thermodynamics,  that  the  rate  of 
radiation  from  a  hot  body  should  be  proportional  to  the  fourth 
power  of  the  absolute  temperature.  Now  this  law  <>f  radiation 
had  been  announced  by  Stefan  as  the  result  of  his  examination 
of  the  experimental  facts  connected  with  radiation;  so  that 
this  fact,  at  least,  supported  Maxwell's  conclusion.  Recently 
.the  Russian  physicist  Lebedew,  and  the  American  physicists 
E.  F.  Nichols  and  Hull,  have  shown,  by  direct  observation, 
that  light  does  exert  a  pressure  such  as  the  theory  describes, 


338  THE   ELECTRIC   CURRENT. 

and  have  measured  its  magnitude  and  found  it  to  conform  pre- 
cisely to  that  predicted  by  the  theory. 

212.  Magnetic  Effect  of  Electric  Convection, — The  concep- 
tion that  an  electric  current  consists  of  a  stream  of  electricity 
along  the  conductor  leads  to  the  hypothesis  that  an  electric 
charge,  when  carried  rapidly  along,  will  set  up  a  magnetic 
field.  This  hypothesis  was  confirmed  by  the  experiments  of 
Kowland,  who  observed  a  magnetic  field  in  the  neighborhood  of 
a  charged  disk  kept  in  rapid  rotation.  Variants  of  Rowland's 
experiment,  executed  by  other  observers,  have  yielded  similar 
results. 


KLKOTRIC   IHSCHAKGE. 


ELECTRIC  DISCHARGE. 

213.  The  Electric  Spark. — A  critical  examination  of  the 
spark  passing  between  oppositely  charged  bodies  shows  cer- 
tain peculiarities  in  it,  which  of  themselves  would  lead  to 
some  modification  of  the  theory  of  electricity  which  has 
hitherto  been  presented.  As  the  discharge  by  means  of  the 
spark  is  variously  modified,  these  peculiarities  become  more 
and  more  prominent,  and  the  necessity  of  some  development  cf 
the  theory  of  electricity  becomes  more  and  more  apparent. 
The  most  important  advance  that  has  been  made  in  physical 
science  in  recent  years  began  with  the  study  of  these  pecu- 
liarities of  the  electric  discharge. 

The  spark,  as  ordinarily  seen,  is  a  bright  line  or  band  cf 
light,  nearly  straight  when  it  is  short,  but  broken  or  zig-zag 
if  its  length  is  greater  than  two  or  three  centimetres.  By 
observations  of  the  spark  in  a  rapidly  revolving  mirror.  Fed- 
dersen  showed  that,  in  ordinary  circumstances,  it  consists  of 
a  succession  of  sparks  occurring  at  regular  intervals,  and 
gradually  diminishing  in  intensity.  The  whole  spark,  how- 
ever, lasts  for  so  short  a  time  that,  for  many  purposes,  we 
may  consider  it  instantaneous.  The  special  peculiarity  of  the 
spark  which  was  referred  to  in  the  preceding  paragraph  is  that 
the  two  ends  of  it,  when  carefully  examined,  do  not  seem  to  be 
exactly  alike.  There  is,  in  fact,  a  characteristic  appearance  at 
the  positive  electrode  which  differs  from  that  at  the  negative 
electrode.  This  difference  between  the  two  ends  of  the  spark 
becomes  more  apparent  when  the  discharge  is  modified,  by 
proper  manipulation,  into  what  is  known  as  the  brush  dis- 
charge. The  brush  discharge  occurs  between  electrodes  which 
are  at  a  considerable  distance,  perhaps  20  centimetres  or  so, 
from  each  other.  When  it  is  established,  a  narrow  band  or 
trunk  of  light  originates  at  the  positive  electrode,  and  branches 
out  into  innumerable  fine  lines  of  light,  which  cease  to  be  visi- 
ble before  they  reach  the  negative  electrode.  At  the  negative 


340  ELECTRIC    DISCHARGE. 

electrode  there  may  generally  be  seen  one  or  more  short 
brushes  of  light  extending  from  the  electrode  for  not  more 
than  a  centimetre.  Between  these  and  the  positive  brush  there 
is  no  visible  evidence  of  the  discharge  at  all.  To  ordinary 
observation,  at  least,  the  brush  discharge  seems  to  be  continu 
ous. 

By  suitable  manipulation  the  discharge  may  be  made  to 
take  still  another  form,  called  the  glow  discharge,  in  which 
the  characteristic  difference  between  the  positive  and  negative 
ends  of  the  discharge  is  also  apparent.  In  this  form  of  the 
discharge,  a  faint  luminous  glow  appears  in  a  thin  layer  ovei 
the  positive  electrode.  Negative  brushes,  similar  to  those 
already  described,  generally  appear  at  the  other  electrode 
These  may  sometimes  be  made  to  coalesce  so  as  to  form  what 
is  called  the  negative  glow.  Even  when  this  is  done,  how- 
ever, the  appearance  of  the  negative  glow  is  distinctly  different 
from  that  of  the  positive  one.  Except  for  these  glows,  no 
light  is  produced  by  the  discharge.  The  discharge  in  this  case 
also  seems  to  be  continuous. 

In  all  our  previoiis  discussions  we  have  treated  the  vitreoue 
and  resinous  electricities,  or  the  positive  and  negative  electric 
conditions,  as  if  they  were  the  precise  counterparts  of  each 
other,  and  differed  in  their  properties  only  by  being  opposite  tc 
each  other.  In  these  various  forms  of  discharge  we  have  found 
differences  between  the  conditions  at  the  two  electrodes  which 
indicate  that  the  differences  between  the  two  kinds  of  eleo 
tricity  may  involve  other  features  than  those  which  have  so 
far  been  assumed.  We  shall  find  that  this  conclusion  is  con 
firmed,  and  our  knowledge  of  the  peculiar  characteristics  oi 
the  two  kinds  of  electricity  very  much  enlarged,  by  the  fur- 
ther study  of  the  discharge. 

214.  Discharge  in  Low  Vacua. — For  the  study  of  the  dia 
charge  in  rarefied  gases  we  use  what  is  called  a  vacuum  tube, 
that  is,  a  glass  tube  or  bulb  from  which  the  air  or  other  gas  oi 
vapor  which  has  filled  it  can  be  withdrawn  by  means  of  an  ail 
pump,  and  which  is  furnished  with  two  terminals  or  electrodes 
supported  on  platinum  wires,  generally  sealed  into  the  walls  of 


ELECTRIC    IJI*CHAR<JE.  341 

the  tube.  In  the  future  we  shall  call  the  electrode  which  is 
joined  to  the  positive  pole  of  the  machine,  the  anode,  and  that 
joined  to  the  negative  pole  of  the  machine,  the  cathode.  When 
the  machine  is  in  operation  and  the  air  is  gradually  with- 
drawn from  the  tube.,  the  discharge,  which  is  at  first  an  in- 
termittent spark,  becomes  more  and  more  frequent  as  the 
racuum  improves,  until  it  is  appreciably  continuous.  In  this 
3ondition  almost  the  whole  of  the  interior  of  the  tube  becomes 
luminous  with  a  faint  rosy  light.  On  examining  the  two 
slectrodes  we  perceive  very  characteristic  differences  in  the 
discharge  around  them.  It  is  not  necessary  to  describe  these 
differences  in  detail.  It  is  sufficient  to  say  that  the  anode  is 
generally  covered  with  a  sheath  or  layer  of  rosy  light  appar- 
ently similar  to  that  in  the  body  of  the  tube.  This  light, 
which  nearly  fills  the  tube,  is  called  the  positive  column.  In 
rery  many  cases  it  appears  to  be  broken  into  saucer-shaped 
layers  or  strata  of  light,  separated  by  darker  spaces.  The 
concave  sides  of  these  strata  are  turned  toward  the  anode, 
ind  the  light  around  the  anode  seems  to  be  simply  one  of  these 
strata  of  modified  form.  The  positive  column  terminates  in 
the  tube,  and  is  separated  from  the  cathode  by  a  dark  space. 
The  cathode  is  surrounded  by  a  bluish  glow,  extending  to  a 
?hort  distance  in  all  directions  from  it,  and  separated  from  it 
by  a  non-luminous  region.  The  color  of  the  positive  column, 
and  the  size  and  distinctness  of  the  strata,  depend  largely  upon 
the  gas  or  vapor  which  is  present  in  the  tube.  The  relative 
prominence  of  the  different  features  of  the  discharge  depends 
upon  the  extent  to  which  the  exhaustion  of  the  tube  is  carried. 
As  the  exhaustion  proceeds,  the  positive  column  diminishes  in 
length  and  brightness  and  the  negative  glow  moves  further 
away  from  the  cathode,  thus  increasing  the  width  of  the  dark 
space  around  it. 

215.  Discharge  in  High  Vacua.— In  very  high  vacua,  such 
as  can  be  obtained  only  by  the  use  of  the  best  air  pumps,  the 
region  occupied  by  the  negative  dark  space  is  so  enlarged  as  to 
811  practically  the  whole  of  the  tube.  In  this  condition  it 
may  be  seen  that  this  space  is  not  entirely  dark,  but  is  filled 


<J42  ELECTRIC    DISCHARGE. 

with  a  faint  bluish  light.  This  light  is  indicative  of  the  pres- 
ance  of  the  so-called  cathode  discharge.  The  cathode  dis- 
charge was  discovered,  and  many  of  its  properties  were  in- 
vestigated, by  Crookes,  and  the  tubes  used  for  its  investigation 
are  frequently  called  Crookes'  tubes. 

Wherever  the  cathode  discharge  reaches  the  walls  of  the 
tube,  it  sets  up  a  peculiar  phosphorescent  illumination,  the 
3olor  of  which  depends  on  the  nature  of  the  glass.  In  the  soda 
glass  ordinarily  employed,  the  color  is  a  pale  green.  Similar 
phosphorescence,  with  characteristic  colors,  is  set  up  in  very 
many  natural  minerals,  or  chemical  preparations.  By  the  aid 
jf  a  screen  covered  with  a  suitable  preparation  the  path  of  the 
uathode  discharge  may  be  studied. 

By  the  direct  examination  of  the  light  of  the  cathode  dis- 
charge, as  well  as  by  the  use  of  phosphorescent  screens,  Crookes 
showed  that  the  discharge  proceeds  from  all  parts  of  the 
cathode  in  lines  normal  to  its  surface.  If,  therefore,  the 
cathode  is  a  flat  plate  which  faces  down  the  tube,  a  principal 
part  of  the  cathode  discharge  will  be  a  nearly  cylindrical 
column  lying  around  the  axis  of  the  tube.  If  the  cathode  is 
a  hollow  spherical  cup,  with  its  concave  side  facing  the  tube, 
the  discharge  will  converge  to  the  centre  of  the  sphere,  and 
will  diverge  in  a  cone  after  passing  through  that  centre.  By 
interposing  an  obstacle,  like  a  sheet  of  metal,  in  the  path  of 
the  cathode  discharge  from  a  flat  cathode,  a  sharply  defined 
shadow  is  cast  on  the  phosphorescent  screen  placed  beyond  the 
obstacle. 

Crookes  noticed  that  when  the  cathode  discharge  was  di- 
rected against  light  films  of  glass,  they  were  moved  as  if  a 
blast  of  air  had  fallen  on  them.  He  constructed  various  me- 
ehanisms,  such  as  wheels  furnished  with  vanes  or  paddles,  and 
found  that,  when  the  cathode  discharge  was  directed  against 
them  they  were  set  in  rotation  as  they  would  be  by  a  blast 
of  air.  He  concluded  that  the  cathode  discharge  contains  mat- 
ter moving  with  a  high  velocity.  In  his  view  this  matter  con- 
sisted of  the  molecules  of  gas  still  remaining  in  the  tubes, 


ELECTRIC    DISCHARGE.  343 

which  were  first  negatively  electrified  by  contact  with  the 
cathode,  and  were  then  repelled  from  it. 

Another  experiment  tried  by  Crookes  was  in  harmony  with 
this  view.  In  it  a  thin  piece  of  platinum  foil,  placed  at  the 
centre  of  a  spherical  cathode,  was  made  red  hot  at  the  point 
where  the  cathode  discharge  met  it.  The  heat  thus  developed 
can  of  course  be  explained  by  the  impacts  of  the  moving  mat- 
ter in  the  discharge. 

When  a  magnet  was  brought  near  the  tube,  the  cathode 
discharge  seemed  to  be  affected  by  it.  To  test  this  more  fully, 
Crookes  allowed  the  discharge  from  a  flat  cathode  to  pass 
through  a  narrow  slit  in  a  sheet  of  metal,  and  then  along  a 
phosphorescent  screen,  on  which  it  developed  a  long  narrow 
band  of  light.  This  band  of  light  was  ordinarily  straight,  but 
when  a  magnet  was  brought  near  the  tube  it  was  bent  into  a 
curve.  The  direction  of  curvature  depended  on  the  direction 
of  the  magnetic  force  which  set  it  up,  and  was  such  as  to  in- 
dicate that  the  stream  of  particles  was  negatively  electrified. 

This  last  conclusion  is  of  great  importance.  It  was  sub- 
sequently verified  by  an  experiment  of  Perrin,  which  was  re- 
peated by  J.  J.  Thomson.  In  this  experiment  the  cathode  dis- 
charge was  allowed  to  pass  through  a  small  opening  in  one 
end  of  a  brass  box  contained  in  the  tube  and  joined  with  an 
electroscope.  As  the  discharge  continued,  the  box  became 
more  and  more  negatively  electrified,  as  would  be  the  case  if  it 
were  receiving  continual  accessions  of  negative  charge  in  its 
interior. 

J.  J.  Thomson  succeeded  in  showing  that  the  discharge  was 
also  acted  on  by  an  electric  field  as  if  it  were  a  stream  of  nega- 
tively charged  particles.  This  cannot  be  done  by  a  field  set  up 
between  electrodes  which  are  outside  the  tube  because  the  in- 
duced charges  in  the  walls  of  the  tube  destroy  the  field  within 
it;  but  by  placing  two  flat  electrodes  within  the  tube,  so  that 
the  discharge  could  pass  between  them,  and  by  using  a  very 
highly  exhausted  tube,  so  that  the  residual  gas  in  it  was  not 
enough  to  rapidly  discharge  the  electrodes,  Thomson  was  able 
to  show  that,  when  the  electrodes  were  oppositely  charged  by 


344  ELECTRIC   DISCHAKGK. 

being  joined  to  the  poles  of  a  battery,  the  cathode  discharge 
was  deflected  between  them.  The  sense  of  the  deflection  indi- 
cated that  the  discharge  was  repelled  by  the  negative  electrode. 

It  was  shown  by  Hittorf  that  when  a  magnetic  field  which 
is  uniform  throughout  the  whole  length  of  the  discharge  is  set 
up  around  the  tube,  the  cathode  stream  becomes  the  arc  of  a 
circle.  As  was  shown  by  Schuster,  this  observation  is  consis- 
tent with,  the  hypothesis  that  the  cathode  stream  is  made  up  of 
negatively  charged  particles  moving  with  a  high  constant  veloc- 
ity. The  amount  of  the  deflection  in  a  given  magnetic  field 
depends  upon  the  velocity  of  the  particles,  and  upon  the  ratio 
of  their  masses  to  the  charges  which  they  carry.  By  combining 
the  results  of  his  own  observations  of  the  deflection  in  a  mag- 
netic field  with  those  of  his  observations  in  an  electric  field, 
J.  J.  Thomson  was  able  to  determine  the  velocity  of  the  par- 
ticles in  the  stream,  and  the  ratio  of  their  masses  to  their 
charges.  He  found  that  the  velocity  was  very  great,  in  many 
cases  not  less  than  one-third  the  velocity  of  light.  The  ratio 
of  the  mass  to  the  charge  was  about  one  thousandth  part  of 
the  ratio  of  the  mass  of  the  hydrogen  atom  to  the  ionic  charge 
which  is  associated  with  it  in  electrolysis.  On  the  supposition 
that  the  charge  of  each  particle  in  the  stream  is  equal  to  the 
ionic  charge,  which  seems  from  all  the  phenomena  of  electro- 
lysis to  be  a  natural  unit  of  electricity,  it  would  follow  that 
the  mass  of  the  particle  is  about  the  one  thousandth  part  of 
the  mass  of  the  hydrogen  atom.  This  conclusion  which,  as 
here  stated,  involves  a  certain  element  of  hypothesis,  was  after- 
wards confirmed  by  Thomson  in  several  ways  which  will  subse- 
quently be  described. 

The  effects  produced  by  the  cathode  discharge  were  so  strik- 
ing that  for  a  long  time  they  occupied  everyone's  attention,  so 
that  the  anode  discharge  was  entirely  neglected.  By  a  suitable 
arrangement  of  the  anode,  however,  it  was  found  that  streams 
or  discharges  also  proceed  from  it,  which  can  be  recognized  by 
their  phosphorescent  effects.  The  effects  produced  on  these 
anode  discharges  by  the  magnetic  and  electric  fields  were  exam- 
ined by  W.  Wien  and  by  Ewers,  and  it  was  proved  that  the 


ELECTRIC   DISCHARGE.  346 

velocity  of  the  particles  in  them  is  very  much  less  than  that 
of  the  particles  in  the  cathode  discharge,  and  that  the  ratio  of 
the  mass  of  these  particles  to  their  charges  is  of  the  same  order 
of  magnitude  as  the  ratio  of  the  mass  of  an  atom  of  an  ordi- 
nary metal  to  the  ionic  charge.  These  experiments  proved  at 
the  same  time  that  the  charges  of  these  anode  streams  are 
positive.  It  is  surmised  that  these  streams  consist  of  posi- 
tively charged  atoms  of  metal  torn  off  from  the  electrode. 

216.  Lcnard  Rays. — While  experimenting  with  a  cathode 
discharge  received  upon  an  aluminium  screen,  Hertz  discovered 
that  phosphorescent  effects  could  be  obtained  behind  the  screen. 
Guided  by  this  discovery,  Lenard  constructed  a  vacuum  tube, 
at  one  end  of  which,  opposite  the  flat  cathode,  the  glass  wall 
was  pierced  by  a  small  aperture  or  window,  closed  with  a  sheet 
of  aluminium  foil.  When  the  cathode  discharge  was  directed 
against  this  window,  peculiar  effects  were  obtained  in  the 
region  immediately  around  it,  outside  the  tube.  Among  these 
effects  were  the  production  of  phosphorescence,  and  of  photo- 
graphic action  on  a  photographic  plate.  In  the  experiment 
with  the  photographic  plate,  the  plate  was  screened  from  the 
light  of  the  tube  by  a  covering  of  black  paper,  through  which 
the  action  penetrated.  It  was  found  that  different  bodies, 
placed  on  the  photographic  plate,  were  penetrated  by  the  action 
in  different  degrees,  it  being  in  general  true  that  the  denser 
bodies  were  less  easily  penetrated  than  those  which  were  less 
dense.  The  cause  of  these  actions  is  evidently  the  cathode 
discharge  which  has  passed  through  the  aluminium. 

One  of  Lenard's  most  interesting  experiments  was  made 
with  a  tube  of  peculiar  construction,  which  was  divided  into 
two  parts  by  a  glass  wall,  furnished  with  an  aluminium  win- 
dow. One  of  these  tubes  was  furnished  with  anode  and 
cathode,  and  the  air  was  exhausted  from  it  to  the  particular 
degree  at  which  the  cathode  discharge  is  most  strongly  excited. 
In  the  other  tube,  as  perfect  a  vacuum  was  made  as  could  be 
made.  With  this  arrangement,  the  cathode  stream  which 
passed  through  the  window  proceeded  in  the  second  tube  in  a 
perfect  vacuum.  Lenard  examined  the  deflections  of  the  cath- 


346  ELECTRIC    DISCHARGE. 

ode  stream,  in  the  second  tube,  produced  by  the  magnetic  and 
electric  fields,  and  obtained  results  for  the  velocity  and  for  the 
ratio  of  the  mass  to  the  charge  which  agree  fairly  well  with 
those  of  Thomson. 

At  the  time  Lenard  discovered  these  effects  he  considered 
the  cathode  discharge  to  be  some  sort  of  wave  disturbance  or 
radiance  in  the  ether,  and  spoke  of  the  effects  as  due  to  rays. 
These  supposed  rays  were  therefore  called  Lenard  raya.  On 
the  view  of  the  nature  of  the  cathode  stream  which  we  have 
adopted,  and  to  which  Lenard  finally  came,  they  are  not  rays 
in  any  proper  sense. 

217.  Roentgen  Rays. — A  series  of  additional  effects  pro- 
duced by  the  discharge  in  a  vacuum  tube  was  discovered  in 
18!)6  by  Roentgen.  These  effects  are  ascribed  to  what,  aa  will 
be  seen,  we  may  properly  call  Roentgen  rays.  When  the  cath- 
ode discharge  falls  on  the  wall  of  the  tube,  or  better,  when  it 
is  concentrated  by  a  spherical  cathode  upon  a  block  of  plati- 
num, the  place  upon  which  it  falls  becomes  the  source  of  theae 
rays.  Some  of  them,  at  least,  pass  out  through  the  walla  of 
the  tube,  and  their  effects  may  be  studied  outaide  of  it.  The 
most  noticeable  effect,  and  the  one  by  which  the  rays  were  dis- 
covered, is  that  of  affecting  the  photographic  plate,  and  of  ex- 
citing phosphorescence  in  many  substances.  The  rays  paas 
through  various  substances  which  are  opaque  to  light  to  a  der 
gree  which  seems  to  depend,  other  things  being  equal,  upon  the 
densities  of  those  substances.  The  photographic  plate  can 
therefore  be  exposed  to  the  rays  while  it  is  enclosed  in  a  plate- 
holder.  Judging  from  the  positions  of  the  shadows  cast  on  the 
plate  by  various  objects,  Roentgen  concluded  that  the  action 
travels  from  the  point  of  origin  in  straight  lines;  and,  from 
the  intensity  of  the  photographic  effect  at  different  distances, 
he  concluded  that  the  intensity  of  the  action  is  inversely  aa 
the  square  of  the  distance  from  the  point  of  origin.  In  theae 
respects  the  action  is  comparable  with  light,  and  we  therefore 
designate  as  rays  the  lines  along  which  it  proceeds.  In  no 
other  respects  have  the  Roentgen  rays  been  proved  to  be  the 
same  as  light  rays.  There  is  no  evidence  of  true  reflection, 


ELECTRIC   DISCHARGE.  347 

though  there  is  some  diffuse  reflection.  Neither  is  there  evi- 
dence of  refraction  or  of  polarization.  In  a  recent  experiment 
Haga  and  Wind  have  brought  forward  evidence  which  indi- 
cates a  diffraction,  from  which,  if  it  is  to  be  accepted,  we  may 
conclude  that  the  wave  lengths  in  the  rays  are  not  more  than 
one  five-thousandth  as  long  as  ordinary  light  waves.  The  hypo- 
thesis that  they  are  such  waves  is  compatible  with  what  is 
known  alxjut  them.  Even  if  we  do  not  accept  the  evidence 
offered  to  prove  that  they  are  diffracted,  the  wave  theory  of 
their  nature  is  still  the  most  acceptable  one.  The  waves  are 
probably  set  up  by  the  sudden  stoppage  of  the  negative  charges 
in  the  cathode  stream,  and  are  therefore  rather  a  series  of 
irregular  pulses  than  a  regular  train  of  waves. 

It  -wa.5  shown  by  J.  J.  Thomson  that  when  the  Roentgen 
rays  pass  through  a  mass  of  air  they  render  it  a  conductor. 
According  to  Thomson  the  conducting  power  is  given  to  the 
gas  by  an  ionization  of  its  atoms,  that  is,  by  a  separation  of 
some  of  the  atoms  into  positive  and  negative  portions. 

218.  The  Negative  Electron.— J.  J.  Thomson  undertook  an 
investigation  of  the  properties  of  ionized  air,  in  order  to  deter- 
mine the  magnitude  of  the  charges  carried  by  the  ions.  Ruth- 
erford had  determined,  in  1897,  the  velocity  of  the  ions  in  an 
electric  field  of  definite  strength.  By  ionizing  air  between  two 
electrodes,  using  the  Roentgen  rays  for  that  purpose,  and  by 
measuring  the  current  transmitted  by  them  between  the  elec- 
trodes when  their  difference  of  potential  was  known,  a  quan- 
tity was  obtained  equal  to  the  number  of  the  ions,  multiplied 
by  the  charge  on  each,  multiplied  by  their  velocity.  The  veloc- 
ity being  known  from  Rutherford's  experiment,  the  charge 
could  be  determined  if  the  number  of  the  ions  could  be  deter- 
mined. To  accomplish  this  Thomson  used  a  discovery  of  C.  T. 
R.  Wilson  that  when  dust-free  air,  saturated  with  water  vapor, 
is  ionized,  and  then  suddenly  expanded  to  a  certain  degree,  leas 
than  that  required  to  produce  condensation  in  ordinary  air, 
condensation  of  water  occurs  on  the  negative  ions.  By  observ- 
ing the  rate  at  which  the  fog  thus  formed  settled,  the  size  of 
the  drops  was  calculated,  and  from  an  observation  of  the  quan- 


348  ELECTRIC   DISCHARGE. 

tity  of  water  condensed,  the  number  of  drops  and  so  of  the 
negative  ions  was  determined.  From  the  data  thus  obtained 
it  was  shown  that  the  charge  of  the  negative  ion  is  the  same 
as  that  belonging  to  the  hydrogen  atom,  and  called  the  ionic 
charge. 

It  was  shown  by  Elster  and  Geitel  that  when  ultra-violet 
light  falls  en  a  charged  conductor  in  air,  the  conductor  is 
speedily  discharged  if  its  charge  is  negative,  while  it  is  not 
discharged  if  its  charge  is  positive.  The  effect  seems  to  be  to 
excite  conditions  in  which  negative  charge  is  liberated  from 
the  body,  for  if  the  light  falls  on  an  uncharged  conductor,  it 
will  gradually  become  positively  charged.  This  action  is  best 
obtained  with  clean  zinc. 

It  was  shown  by  the  same  observers  that  the  discharge 
from  a  plate  of  zinc  is  diminished,  and  finally  checked  alto- 
gether, if  the  zinc  is  in  a  sufficiently  strong  magnetic  field,  the 
lines  of  force  of  which  are  parallel  with  its  surface.  The 
experiment  succeeds  only  when  the  zinc  is  in  a  vacuum.  On 
the  supposition  that  the  escape  of  the  negative  charge  takes 
place  by  the  emission  of  negative  ions  from  the  surface, 
Thomson  showed  that  the  effect  of  the  magnetic  field  ought  to 
be  to  change  the  straight  paths  of  the  ions  into  cycloids,  and 
that  if  the  discharge  is  taking  place  between  the  zinc  and  an 
electrode  placed  parallel  with  it  and  near  it,  a  current  ought 
to  pass  between  the  plates,  of  which  the  intensity  will  remain 
appreciably  the  same  so  long  as  the  cycloidal  paths  of  the 
ions  meet  the  second  plate,  but  will  change  abruptly,  or  at 
least  very  rapidly,  for  such  a  strength  of  the  magnetic  field 
that  most  of  the  cycloidal  paths  do  not  meet  the  second  plate. 
By  trial  Thomson  convinced  himself  that  such  an  abrupt 
change  in  the  current  occurred,  and  that  his  hypothesis  was 
therefore  justified.  From  the  distance  between  the  plates,  he 
determined  the  magnitude  of  the  cycloidal  paths,  and  from 
this  and  the  observed  values  of  the  difference  of  potential 
between  the  plates,  and  the  strength  of  the  magnetic  field,  he 
computed  the  ratio  of  the  mass  of  the  ion  to  the  charge  upon 
it.  He  found  this  ratio  to  be  the  same  as  that  which  he  had 


BLKCTRIC   DISCHARGE.  349 

determined  for  the  particles  in  the  cathode  stream.  By  con- 
densing water  drops  on  the  ions,  in  the  way  already  explained, 
he  determined  their  number,  and  hence  determined  the  charge 
on  the  ion,  which  he  found  in  this  case  also  to  be  equal  to  the 
ionic  charge  of  the  hydrogen  atom.  In  this  way,  without  hy- 
pothesis, he  showed  that  the  mass  of  the  negative  ion  which 
is  associated  with  an  ionic  charge  is  about  the  one-thousandth 
part  of  the  mass  of  an  hydrogen  atom.  It  can  hardly  be 
doubted  that  the  negative  charges  in  the  cathode  stream  are 
also  ionic  charges-,  and  that  the  mass  of  the  particle  is  far 
less  than  that  of  the  hydrogen  atom. 

The  negative  ions  thus  recognized  must  be  distinguished 
from  the  ions  in  electrolysis,  which  are  masses  of  matter  con- 
sisting of  one  or  more  atoms,  and  associated  with  a  number 
of  ionic  charges  equal  to  the  valency.  To  denote  this  differ- 
ence, we  may  call  the  negative  ions  of  the  cathode  stream,  or 
those  produced  in  a  gas  by  the  Roentgen  rays,  or  otherwise, 
by  the  name  electrons. 

To  describe  these  results,  Thomson  supposes  that  the  atoms 
of  a  gas  consist  of  a  positive  portion  joined  to  a  number  of 
negative  electrons,  which  neutralize  the  positive  charge,  and 
that,  when  the  gas  is  ionized,  some  of  its  atoms  liberate  each 
one  electron,  and  become  themselves  positively  charged. 

210.  Becquerel  Rays. — Not  long  after  the  discovery  of  the 
Roentgen  rays  the  French  physicist  Becquerel  discovered  that 
similar  effects  to  those  produced  by  them  could  be  obtained 
from  masses  of  uranium  or  of  ores  containing  uranium.  The 
activity  of  the  uranium  was  very  slight,  and  several  days 
were  required  to  obtain  the  photographic  effects  by  which  the 
properties  of  these  rays  were  studied.  They  were  found  to  be 
in  general  similar  to  those  of  the  Roentgen  rays,  but  certain 
differences  were  noticed,  which  could  not  be  fully  studied  on 
account  of  the  feebleness  of  the  action.  The  action,  however, 
was  considered  to  be  due  to  rays  like  the  Roentgen  rays. 

220.  Radium  and  Radioactive  Substances. — M.  and  Mad- 
ame Curie  found  that  certain  uranium-bearing  minerals  were 
more  active  than  metallic  uranium.  As  the  result  of  long 


350  ELECTRIC    DISCHARGE. 

continued  labor,  they  succeeded  in  isolating  from  pitch- 
blende, an  ore  of  uranium,  a  small  quantity  of  a  salt  of  a  new 
element,  which  they  named  radium. 

Thiselement  exhibits  properties  similar  to  those  of  uranium, 
but  to  a  far  greater  degree.  They  are  called  generally  radio- 
active properties.  The  most  delicate  and  ready  way  of  detecting 
radioactivity  is  by  the  use  of  a  sensitive  electroscope,  which 
will  gradually  be  discharged  if  the  air  around  it  is  ionized  by 
the  radioactive  body.  Thorium,  among  the  elements  already 
known,  was  found  also  to  be  radioactive,  and  the  discovery 
of  several  other  new  radioactive  elements  has  been  announced. 
It  is  not  yet  determined  whether  these  other  elements  have  a 
real  existence,  or  whether  the  effects  attributed  to  them  are  not 
rather  due  to  radium  or  to  a  product  of  its  disintegration. 

The  properties  of  thorium  and  radium  have  been  very  care- 
fully studied  by  Rutherford  and  Soddy  and  their  associates. 
The  conclusions  which  they  have  drawn  are  too  complicated 
for  a  brief  description,  but  certain  of  their  results  and  a  gen- 
eral statement  about  the  others  can  be  given.  They  conclude 
that  the  atoms  of  these  bodies  are  one  by  one  disintegrating. 
This  disintegration  is  accomplished  by  sending  off  negative 
electrons,  and  also  positively  charged  bodies.  Only  a  rela- 
tively very  small  number  of  atoms  break  up  at  once,  though 
the  actual  number  is  very  large;  sufficiently  large  for  their 
emission  to  constitute  streams  of  negative  electrons  and  posi- 
tive bodies.  The  stream  of  negative  electrons  can  be  detected 
by  their  behavior  in  a  magnetic  field.  It  is  exactly  like  the 
negative  stream  from  the  cathode.  The  stream  of  positively 
charged  bodies  is  also  affected  by  the  magnetic  field,  in  such  a 
way  as  to  show  that  the  velocities  of  those  bodies  are  much 
less  than  the  velocities  of  the  negative  electrons,  and  that  their 
masses  are  much  greater.  This  conclusion  follows  also  from 
the  fact  that  the  positive  streams  are  intercepted  almost  en- 
tirely by  a  very  slight  obstacle,  like  a  sheet  of  paper,  while 
the  negative  streams  have  much  greater  penetrating  power.  A 
third  set  of  streams  or  rays  has  been  recognized  which  can 
penetrate  through  several  inches  of  iron  or  lead.  In  the  case 


ELECTRIC    DISCHARGE.  361 

of  radium  another  set  of  bodies,  which  Rutherford  calls 
the  emanation,  leaves  the  mass.  This  emanation  seems  to  be 
the  remnants  of  the  radium  atoms,  after  they  have  undergone 
the  first  disintegration.  The  emanation  undergoes  another 
disintegration,  during  which  negative  electrons  and  positively 
charged  bodies  are  again  emitted.  The  experiments  of  Ram- 
say and  Soddy  indicate  that  one  of  the  products  of  these  dis- 
integrations, probably  the  positively  charged  bodies  first 
emitted,  are  atoms  of  the  element  helium. 

Rutherford  and  Soddy  proved  with  thorium  that  it  was 
possible  to  separate  it  by  chemical  processes  into  two  parts, 
one  of  which  emitted  negative  electrons  and  positively  charged 
bodies,  while  the  other  emitted  only  the  latter.  With  lapse  of 
time  both  portions  acquired  the  normal  type  of  radioactivity. 
We  can  explain  these  results  only  by  supposing  that  a  contin- 
ual alteration  is  going  on  in  the  thorium  of  one  class  of  atoms 
into  another,  and  that  the  atoms  thus  formed  are  continually 
breaking  down  into  the  products  which  indicate  radioactivity. 
The  normal  radioactivity  of  the  thorium  is  that  which  is 
reached  when  as  many  active  atoms  are  being  produced  as  are 
being  destroyed.  Not  only  in  the  case  of  thorium,  but  in  the 
case  of  the  other  radioactive  elements,  especially  of  radium, 
observation  indicates  a  gradual  disintegration  and  destruction 
of  these  elements.  Rutherford  calculates  that  the  life  of  a 
mass  of  radium  is  about  two  thousand  years.  Tha*  of  thorium, 
which  disintegrates  much  less  rapidly,  is  many  millions  of 
years. 

Radioactive  properties,  which  have  been  shown  to  be  simi- 
lar to  those  of  the  emanation  of  radium,  are  exhibited  by 
specimens  of  water  drawn  from  wells  in  all  parts  of  the  world. 
We  may  conclude  from  this  observation  that  radioactive  bodies 
are  very  widely  distributed  in  the  earth's  crust. 

221.  Electric  Theories  of  Matter. — Kaufmann  has  exam- 
ined by  experiment  the  effect  produced  by  a  magnetic  field  on 
those  rays  from  radium  which  correspond  to  the  cathode 
stream.  He  finds  that  under  the  action  of  the  field  the  stream 
spreads  out  into  a  gradually  widening  band.  By  studying  the 


352  ELECTRIC   DI8CHARQK. 

curvatures  of  different  parts  of  the  stream,  he  finds  that  they 
are  consistent  with  the  supposition  that  the  electrons  in  the 
stream  have  no  mass.  It  was  shown  by  J.  J.  Thomson  from 
Maxwell's  theory  of  electricity  that  when  a  charged  sphere  is 
moving  rapidly  through  the  ether,  and  is  acted  on  by  a  force 
tending  to  change  its  motion,  it  will  behave  as  if  its  masa 
were  greater  than  the  mass  of  the  sphere,  by  an  amount  which 
depends  on  the  electric  charge  upon  it,  and  the  velocity  with 
which  it  is  moving.  This  general  conclusion,  that  a  moving 
charge  will  have  an  apparent  inertia,  due  to  its  interaction 
with  the  ether,  has  been  confirmed  by  other  investigators,  and 
especially  by  Abraham,  with  whose  equations  Kaufmann  inter- 
preted his  results.  Kaufmann  finds  that  the  paths  of  the 
electrons  sent  out  by  the  radium  are  such  as  to  indicate  that 
the  mass  of  the  electron  depends  upon  its  velocity,  and  in  such 
a  way  as  it  would  if  the  electron  possessed  no  mass  except 
the  apparent  mass  due  to  its  electric  charge.  If  this  result  is 
accepted,  we  must  conclude  that  the  electron  is  a  small  por- 
tion of  electricity,  separate  from  matter. 

The  apparent  mass  which  a  moving  electric  charge  can  have 
indicates  a  theory  of  matter  in  which  the  material  atoms  are 
assumed  to  be  assemblages  of  electrons,  moving  around  each 
other  in  very  small  orbits,  but  with  enormous  velocity.  The 
ether  is  assumed  to  be  a  continuous  medium,  possessing  a 
peculiar  elastic  reaction  to  torques  or  twists.  Theories  rest- 
ing on  this  general  supposition  have  been  developed  by  H.  A. 
Lorentz  and  by  Larmor,  and  have  been  shown  capable  of  ac- 
counting for  most,  if  not  all  of  the  phenomena  of  light  and 
electricity,  in  their  relations  to  moving,  as  well  as  to  station- 
ary bodies. 

On  this  general  theory  a  radioactive  body  is  one  in  which 
the  electrons  in  the  atom  are  so  numerous  and  are  moving  in 
such  manner  that  the  bonds  which  unite  them  together  are  not 
always  stable.  This  being  so,  if  the  limit  of  stability  is  passed, 
Uhe  atom  may  break  up,  one  of  the  results  of  its  breaking  up 
being  generally  the  emission  of  at  least  one  electron. 


PLAT 


Rig.  2 


i 


PLATE  5 


Fig.  42 


Fig.  43 


Fig.  44 


Fig-  45 


PLATE  6 


Fig.  46 


F'&-  47 


PL 


ATE    7 


Fig-  55 


Fig.  56 


Fig-  59 


Fig.  60 


PLATE  8 


Fig.  62 


FiK.  68 


Fig.  69 


This  book  is  DUE  on  the  last  date  stamped  bel( 

I 

NOV  2  8  1994 
OCT  i'i 

QCT  23  1940 

A*UG77     1942 
DEC  2 1  1^42 

NOV  l  4  1948 
MAR  2  8-195Q 


Form  L-9-15wi-7,'32 


QC31 

M27s     Magie   - 

Syllabus__of_ 
a  course   of 
lectures   on 

physicsT" 


Q.CI3 


UMIVERSITY  of  CALIFORNIA 


<  TBRABY 


